Straight Lines Slides-239
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Transcript of Straight Lines Slides-239
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Straight Line MC Sir
1.Basic Geometry, H/G/O/I 2.Distance & Section formula, Area of triangle,
co linearity3.Locus, Straight line3.Locus, Straight line4.Different forms of straight line equation5.Examples based on different form of straight
line equation6.Position of point with respect to line, Length
of perpendicular, Angle between two straight lines
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7.Parametric Form of Line
8.Family of Lines
Straight Line MC Sir
9.Shifting of origin, Rotation of axes
10.Angle bisector with examples
11.Pair of straight line, Homogenization
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Straight Line MC Sir
No. of Questions
2008 2009 2010 2011 2012
2 1 1 2 --
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Basic ConceptsBasic Concepts
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Determinants
Array of No.
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
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Value of x and y In Determinant formIn Determinant form
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Method of Solving 2 × 2 Determinant
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Method of Solving 2 × 2 Determinant
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Method of Solving 3 × 3 Determinant
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Definition of Minor[M ij]
Minor of an element is defined as minor
determinant obtained by deleting a particular
row or column in which that element lies.
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Example
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Example
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Cofactor
Cij = (-1)i + j M ijCij = (-1) M ij
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Example
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Example
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Value of Determinant in term of Minor and Cofactor
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Value of Determinant in term of Minor and Cofactor
D = a11 M11 – a12 M12 + a13 M13D = a11 M11 – a12 M12 + a13 M13
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Value of Determinant in term of Minor and Cofactor
D = a11 M11 – a12 M12 + a13 M13D = a11 M11 – a12 M12 + a13 M13
D = a11 C11 + a12 C12 + a13 C13
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Note
A determinant of order 3 will have 9 minors
each minor will be a minor of order 2
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Note
A determinant of order 3 will have 9 minors
each minor will be a minor of order 2
A determinantof order 4 will have16 minorsA determinantof order 4 will have16 minors
each minor will be a minor of order 3
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Note
A determinant of order 3 will have 9 minors
each minor will be a minor of order 2
A determinantof order 4 will have16 minorsA determinantof order 4 will have16 minors
each minor will be a minor of order 3
We can expand a determinant in 6 ways (for
3×3 determinant)
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Examples
Q.
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Examples
Q.
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Examples
Q.
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Examples
Q.
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Properties of DeterminantProperties of Determinant
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The value of determinant remains same ifrow and column are interchanged
P-1 Property
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The value of determinant remains same ifrow and column are interchanged
P-1 Property
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Skew Symmetric Determinant
DT = - DD = - DValue of skew symmetric determinant is zero
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P-2 Property
If any two rows or column be interchanged thevalue of determinant is changed in sign only.
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P-2 Property
If any two rows or column be interchanged thevalue of determinant is changed in sign only.Example :
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P-3 Property
If a determinant has any two rowor columnsame then its value is zero.
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P-3 Property
If a determinant has any two rowor columnsame then its value is zero.Example :
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P-4 Property
If all element of any row or column bemultiplied by same number than determinant ismultiplied by that number
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P-4 Property
If all element of any row or column bemultiplied by same number than determinant ismultiplied by that number
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P-5 Property
If each element of any rowor column can beexpressed as sumof two terms thendeterminant can be expressed as sumof twodeterminants.determinants.
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P-5 Property
If each element of any rowor column can beexpressed as sumof two terms thendeterminant can be expressed as sumof twodeterminants.determinants.Example :
Result can be generalized.
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Example
Q. If
Find thevalueofFind thevalueof
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P-6 Property
The value of determinant is not changed by adding
to the element of any rowor column the same
multiplesof thecorrespondingelementsof anyothermultiplesof thecorrespondingelementsof anyother
row or column.
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Example
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Example
D = D'
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Remainder Theorem
Any polynomial P(x) when divided by (x -α)
then remainder will be P(α)
If P (α) = 0⇒ x - α is factor of P (x)
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P-7 Property
If by putting x = a the value of determinant
vanishes then (x–a) is a factor of the
determinant.
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(i) Create zeros
Method
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(i) Create zeros
(ii) Take common out of rows and columns
Method
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(i) Create zeros
(ii) Take common out of rows and columns
(iii) Switch over between variable to create
Method
(iii) Switch over between variable to create
identical rowor column.
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ExampleShowthat
= (x – y) (y – z) (z – x)
Q.
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Prove thatQ.
= (a2 + b2 + c2) (a + b + c) (a – b) (b – c) (c – a)
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Prove thatQ.
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Find f(100) (JEE99)
Q.
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Q.
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Q.
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Q.
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Q.
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Q.
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System of Equation
a1x + b
1y + c
1= 0
a x + b y + c = 0a2x + b
2y + c
2= 0
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System Consistant
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System Consistant
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System Consistant
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TriangleA
bc b
C
c
B a
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TriangleA
bc b
C
c
B a
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Angle bisector
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Angle bisector
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Angle bisector
I is called Incentre (Point of concurrency ofinternal angle bisector)
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Incircle
Circle who touches sides of triangle is called
incircle,
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Altitude
Perpendicular fromvertex to opposite side
(Orthocenter)
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Median
Line joining vertex to mid point of opposite sides
(Centroid)
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Perpendicular bisector
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Perpendicular bisector
Any point on perpendicular bisector is at equal
distance fromA & B
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Circumcircle
O is circumcentre
R is circumradius
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Note
In Right angle triangle
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NoteG (centroid) & I (Incentre) always lies in interior
of triangle whereas H (Orthocenter) & O
(Circumcentre) lies inside, outside or periphery
depending upon triangle being acute, obtuse or
right angle.
H G O
2 : 1
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Quadrilaterals
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Note
Sumof all interior angles of n sided figure is
= (n – 2)π
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Parallelogram
(i) Opposite sides are parallel & equal
(ii) adjacent angles are supple-
-mentary-mentary
(iii) Diagonals are bisected.
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Parallelogram
Area parallelogram= d1
d2
sin φ
DE = b sinθ
Area of parallelogram= ab sinθ
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Rhombus
Parallelogramwill be Rhombus
If
(i) Diagonal are perpendicular
(ii) Sides equal
(iii) Diagonal bisects angle of parallelogram
(iv) Area of Rhombus =
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RectangleParallelogramwill be Rectangle
If
(1) Angle 90°
(2) Diagonals are equal
(3) a2 + b2 = c2
Rectangle is cyclic quadrilateral
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SquareA Rectangle will be square
If
(1) Sides equal
(2) Diagonal are perpendicular
(3) Diagonal are angle bisector
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Square
A Rhombus will be square
If
(1) Diagonal are equal
(2) Angle 90°
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Trapezium
(1) One pair of opposite sides are parallel
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Kite
(1) One diagonal divide figure into two congruentpart
(2) Diagonalareperpendicular(2) Diagonalareperpendicular
(3) Area = d1
d2
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Cyclic Quadrilateral
i. Vertices lie on circle
ii. A + C = π = B + D
iii. AE × EC= BE × DE
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Note
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Note
(EB) (EA) = (EC) (ED) = (ET)2 = (EP) (EQ)
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PTolmey’s Theorem
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PTolmey’s Theorem
Sum of product of opposite side = Product ofdiagonals
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PTolmey’s Theorem
Sum of product of opposite side = Product ofdiagonals
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PTolmey’s Theorem
Sum of product of opposite side = Product ofdiagonals
(AB) (CD) + (BC) (AD) = (AC) (BD)
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(Point) Geometry
Co-
ordinate
(x) Algebra
ordinate
Geometry
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Distance Formulae
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Distance Formulae
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Distance Formulae
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ExampleFind distance between following points :-
Q.1 (1, 3), (4, -1)
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Find distance between following points :-
Q.2 (0, 0), (-5, -12)
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Find distance between following points :-
Q.3 (1,1), (16, 9)
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Find distance between following points :-
Q.4 (0, 0), (40, 9)
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Find distance between following points :-
Q.5 (0, 0) (2cosθ, 2sinθ)
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Section Formulae(Internal Division)
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Section Formulae(Internal Division)
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Section Formulae(Internal Division)
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Coordinate of mid point of A (x
1, y
1), B (x
2, y
2)
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Coordinate of mid point of A (x
1, y
1), B (x
2, y
2)
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ExampleQ. Find points of trisection of (1, 1) & (10, 13)
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Co-ordinate of G
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Co-ordinate of G
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Co-ordinate of G
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ExampleQ. Find mid points of sides of∆ if vertices are
given (0, 0), (2, 3), (4, 0). Also find coordinate
of G
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Q. Find the ratio in which point on x axis divides
the two points. (1,1), (3, -1) internally.
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Section Formulae(External Division)
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Section Formulae(External Division)
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Section Formulae(External Division)
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ExampleQ. Find the point dividing (2, 3), (7, 9) externally
in the ratio 2 : 3
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Harmonic Conjugate
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Harmonic ConjugateIf a point P divides ABinternally in the ratio a : band point Q divides ABexternally in the ratio a : b,then P & Q are said to be harmonic conjugate ofeachotherw.r.t. ABeachotherw.r.t. AB
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Harmonic ConjugateIf a point P divides ABinternally in the ratio a : band point Q divides ABexternally in the ratio a : b,then P & Q are said to be harmonic conjugate ofeachotherw.r.t. ABeachotherw.r.t. AB
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Harmonic ConjugateIf a point P divides ABinternally in the ratio a : band point Q divides ABexternally in the ratio a : b,then P & Q are said to be harmonic conjugate ofeachotherw.r.t. ABeachotherw.r.t. AB
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Harmonic Mean
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Harmonic Mean
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(i) Internal & external bisector of an angle of a
∆ divide base harmonically
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(i) Internal & external bisector of an angle of a
∆ divide base harmonically
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(i) Internal & external bisector of an angle of a
∆ divide base harmonically
D & D' are harmonic conjugate of each other
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ExampleQ. If coordinate of A & B is (0, 0) & (9, 0) find
point which divide ABexternally in the ratio1 : 2 find its harmonic conjugate.
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External & Internal common tangents dividesline joining centre of two circles externally &internally at the ratio of radii
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External & Internal common tangents dividesline joining centre of two circles externally &internally at the ratio of radii
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External & Internal common tangents dividesline joining centre of two circles externally &internally at the ratio of radii
O1
and O2
are harmonic conjugate each other.
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Co-ordinates of Incentre (I)
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Co-ordinates of Incentre (I)
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Co-ordinates of Incentre (I)
a
bc
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Q.1 If P (1, 2), Q (4, 6), R (5, 7) and S (a, b) are thevertices of parallelogramPQRSthen(A) a = 2, b = 4 (B) a = 3, b = 4(C) a = 2, b = 3 (D) a = 1 or b = -1
[IIT-JEE 1998]
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Q.2 The incentre of triangle with vertices ,(0, 0) and (2, 0) is(A) (B)(C) (D)
[IIT-JEE 2000]
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S.L. Loney
Assignment - 1
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Find the distance between the following pairs ofpointsQ.1 (2, 3) and (5, 7)Q.2 (4, -7) and (-1, 5)Q.3 (a, 0) and (0, b)Q.4 (b + c, c + a) and (c + a, a + b)Q.5 (acosα, a sinα) and(acosβ, asinβ)Q.5 (acosα, a sinα) and(acosβ, asinβ)Q.6 (am
12, 2am
1) and (am
22, 2am
2)
Q.7 Lay down in a figure the position of the points(1, -3) and (-2, 1), and prove that the distancebetween themis 5.
Q.8 Find the value of x1
if the distance between thepoints (x
1, 2) and (3, 4) be 8.
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Q.9 A line is of length 10 and one end is at thepoint (2, -3); if the abscissa of the other endbe 10, prove that its ordinate must be 3 or -9.
Q.10Prove that the points (2a, 4a), (2a, 6a) andare the vertices of an equilateral
trianglewhosesideis 2a.trianglewhosesideis 2a.
Q.11 Prove that the points (2, -1), (1, 0), (4, 3), and(1, 2) are at the vertices of a parallelogram.
Q.12Prove that the points (2, -2), (8,4), (5,7)and (-1,1) are at the angular points of arectangle.
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Q.13 Prove that the point is the centre ofthe circle circumscribing the triangle whoseangular points are (1, 1), (2, 3), and (-2, 2).Find the coordinates of the point which
Q.14 Divide the line joining the points (1, 3) and(2, 7) in theratio3 : 4.(2, 7) in theratio3 : 4.
Q.15 Divides the same line in the ratio 3 : -4.Q.16 Divides, internally and externally, the line
joining (-1, 2) to (4, -5) in the ratio 2 : 3.Q.17 Divide, internally and externally, the line
joining (-3, -4) to (-8, 7) in the ratio 7 : 5
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Q.18 The line joining the point (1, -2) and (-3, 4) istrisected; find the coordinate of the points oftrisection.
Q.19 The line joining the points (-6, 8) and (8, -6) isdivided into four equal pats; find thedivided into four equal pats; find thecoordinates of the points of section.
Q.20 Find the coordinates of the points whichdivide, internally and externally, the linejoining the point (a + b, a – b) to the point(a – b, a + b) in the ratio a : b.
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Q.21 The coordinates of the vertices of a triangleare (x
1y
1), (x
2, y
2) and (x
3, y
3). The line
joining the first two is divided in the ratio l :k, and the line joining this point of divisionto the opposite angular point is then dividedin the ratio m : k + l. Find the coordinateofin the ratio m : k + l. Find the coordinateofthe latter point of section
Q.22 Prove that the coordinate, x and y of themiddle point of the line joining the point (2,3) to the points (3, 4) satisfy the equation,
x – y + 1 = 0
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Coordinates of I1, I
2& I
3
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Coordinates of I1, I
2& I
3
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Coordinates of I1, I
2& I
3
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Coordinates of I1, I
2& I
3
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Coordinates of I1, I
2& I
3
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ExamplesQ.1 Mid point of sides of triangle are (1, 2), (0, -1)
and (2, -1). Find coordinate of vertices
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Q.2 Co-ordinate A, B, C are (4, 1), (5, -2) and (3, 7)Find D so that A, B, C, D is ||gm
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Q.3 Line 3x + 4y = 12, x = 0, y = 0 forma ∆.Find the centre and radius of circles touchingthe line & the co-ordinate axis.
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Q.4 Orthocenter and circumcenter of a∆ABC are(a, b), (c, d). If the co-ordinate of the vertex Aare (x
1, y
1) then find co-ordinate of middle
point of BC.
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Q.5 Vertices of a triangle are (2, -2), (-2, 1), (5, 2).Find distance between circumcentre & centroid.
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Area of equilateral triangle
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Area of Triangle
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Area of Triangle
Where (x1, y
1) (x
2, y
2), (x
3, y
3) are coordinates of
vertices of triangle
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Condition of collinearity ofA (x
1, y
1), B (x
2, y
2), C(x
3, y
3)
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Condition of collinearity ofA (x
1, y
1), B (x
2, y
2), C(x
3, y
3)
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Area of n sided figure
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Area of n sided figure
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ExampleQ.1 Find k for which points (k+ 1, 2 – k),
(1 – k, –k) (2 + K, 3 – K) are collinear.
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Q.2 If points (a, 0), (0, b) and (1, 1) are collinear
then prove that
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Q.3 Find relation between x & y if x, y lies on linejoining the points (2,–3) and (1, 4)
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Q.4 Showthat (b, c + a) (c, a + b) and (a, b + c) arecollinear.
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Q.5 If the area of∆ formed by points (1, 2), (2, 3)and (x, 4) is 40 sq. unit. Find x.
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Q.6 Find area of quadrilateral A (1, 1); B (3, 4);C (5, -2) and D (4, -7) in order are thevertices of a quadrilateral.
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Q.7 Find co-ordinate of point P if PA= PB andarea of∆PAB = 10 if coordinates of A and Bare (3, 0) and (7,0) respectively.
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Q.8 Find the area of the∆ if the coordinate ofvertices of triangle are
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Assignment - 2Assignment - 2
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Find the areas of the triangles the coordinate of
whose angular points are respectively.
Q.1 (1, 3), (-7, 6) and (5, -1)
Q.2 (0, 4), (3, 6) and (-8, -2).
Q.3 (5, 2), (-9, -3) and (-3, -5)
Q.4 (a, b + c), (a, b – c) and (-a, c)
Q.5 (a, c + a), (a, c) and (-a, c – a)
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Prove (by shewing that the area of the triangle
formed by themis zero that the following sets of
three points are in a straight line :
Q.6 (1, 4), (3, -2) and(-3, 16)Q.6 (1, 4), (3, -2) and(-3, 16)
Q.7 , (-5, 6) and (-8, 8).
Q.8 (a, b + c), (b, c + a), and (c, a + b)
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Find the area of the quadrilaterals the coordinates
of whose angular points, taken in order, are :
Q.9 (1, 1), (3, 4), (5, -2) and (4, -7)
Q.10 (-1, 6), (-3, -9), (5, -8) and (3, 9)
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LOCUSLOCUS
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To Find Locus(1) Write geometrical condition & convert themin
algebraic
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To Find Locus(1) Write geometrical condition & convert themin
algebraic
(2) Eliminate variable
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To Find Locus(1) Write geometrical condition & convert themin
algebraic
(2) Eliminate variable
(3) Get relation between h and k.
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To Find Locus(1) Write geometrical condition & convert themin
algebraic
(2) Eliminate variable
(3) Get relation between h and k.
(4) To get equation of locus replace h by x & k by y
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ExampleQ.1 Find locus of curve / point which is equidistant
from point (0, 0) and (2, 0)
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Q.2 If A (0, 0), B (2, 0) find locus of point P such
that∠APB = 90°
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Q.3 If A (0, 0), B (2, 0) find locus of point P such
that area (∆ APB) = 4
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Q.4 If A & B are variable point on x and y axis
such that length (AB) = 4. Find :
(i) Locus of mid point of AB
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Q.4 If A & B are variable point on x and y axis
such that length (AB) = 4. Find :
(ii) Locus of circumcentre of∆AOB
![Page 179: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/179.jpg)
Q.4 If A & B are variable point on x and y axis
such that length (AB) = 4. Find :
(iii) Locus of G of∆AOB
![Page 180: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/180.jpg)
Q.4 If A & B are variable point on x and y axis
such that length (AB) = 4. Find :
(iv) Find locus of point which divides
segment ABinternally in the ratio 1 : 2, 1
from x axis
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Q.5 A(1, 2) is a fixed point. A variable point B lies
on a curve whose equation is x2+y2 = 4. Find
the locus of the mid point of AB.
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Parametric pointParametric point
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ExampleQ.1 Find equation of curve represented
parametrically by x = cosθ, y = sinθ
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Q.2 Find equation of curve if x = 2cosθ, y = sinθ
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Q.3 Find equation of curve if x = secθ, y = 2tanθ
![Page 186: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/186.jpg)
Q.4 Find equation of curve if x = at2, y = 2at
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Q.5 Find locus of point P such that ;PF
1+ PF
2= 2a & F
1≡ (c, 0) & F
2≡ (-c, 0)
![Page 188: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/188.jpg)
Q.6 Find locus of point P such that|PA– PB| = 2a & coordinates of A, B are(c, 0) & (-c, 0)
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Assignment - 3Assignment - 3
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Sketch the loci of the following equations :
Q.1 2x + 3y = 10
Q.2 4x – y = 7
Q.3 x2 – 2ax + y2 = 0
Q.4 x2 – 4ax+ y2 + 3a2 = 0Q.4 x2 – 4ax+ y2 + 3a2 = 0
Q.5 y2 = x
Q.6 3x = y2 - 9
![Page 191: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/191.jpg)
A and B being the fixed points (a, 0) and (-a, 0)
respectively, obtain the equations giving the locus
of P, when :
Q.7 PA2 – PB2 = a constant quantity = 2k2
Q.8 PA= nPB,n being constant.
Q.9 PB2 + PC2 = 2PA2, C being the point (c, 0)
Q.10 Find the locus of a point whose distancefrom the point (1, 2) is equal to its distancefrom the axis of y.
![Page 192: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/192.jpg)
Find the equation to the locus of a point which is
always equidistant from the points whose
coordinate are
Q.11 (1, 0) and(0, -2)Q.11 (1, 0) and(0, -2)
Q.12 (2, 3) and (4, 5)
Q.13 (a + b, a – c) and (a – b, a + b)
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Find the equation to the locus of a point which
moves so that
Q.14 Its distance fromthe axis of x is three timesits distance fromthe axis of y.
Q.15 Its distance fromthe point (a, 0) is alwaysfour times its distance fromthe axis of y.
Q.16 The sumof the squares of its distances fromthe axes is equal to 3.
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Q.17 The square of its distance fromthe point (0, 2)
is equal to 4.
Q.18 Its distance fromthe point (3, 0) is three times
its distance from(0, 2)
Q.19 Its distance fromthe axis of x is always one
half its distance fromthe origin.
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Straight Line
Locus of point such that if any two point of thislocus are joined they define a unique direction.
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Inclination of Line
![Page 197: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/197.jpg)
Inclination of Line
![Page 198: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/198.jpg)
Inclination of Line
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Slope / Gradient (m)
![Page 200: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/200.jpg)
Slope / Gradient (m)
m = tanα ; α ≠ π/2
![Page 201: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/201.jpg)
Slope of line joining two points
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Slope of line joining two points
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Slope of line joining two points
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ExampleQ.1 Find slope of joining points (1, 1) & (100, 100)
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Q.2 Find slope of joining points (1, 0) & (2, 0)
![Page 206: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/206.jpg)
Q.3 Find slope of joining points (1, 9) & (7, 0)
![Page 207: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/207.jpg)
Equation of Line inVarious FormVarious Form
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General Form
ax + by + c = 0ax + by + c = 0
![Page 209: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/209.jpg)
Point Slope Form
(y – y1) = m (x – x
1)
1 1
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Q.1 Find equation of line having slope 2 and
passing through point (1, 3)
Example
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Q.2 Find equation line having slope and passing
through point (1, 7)
![Page 212: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/212.jpg)
Q.3 Line passing through (1, 0) and (2, 1) is rotated
about point (1,0) by an angle 15° in clockwise
direction. Find equation of line in newposition.
![Page 213: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/213.jpg)
Two Point Form
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Two Point Form
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Q.1 Find equation of line joining (1, 1), (3, 4)
Example
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Determinant Form
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Determinant Form
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Intercept Slop Form
![Page 219: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/219.jpg)
y = mx + c
Intercept Slop Form
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y = mx + c
y = mx [if line passes through origin]
Intercept Slop Form
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Q.1 Find slope, x intercept, y intercept of lines(i) 2y = 3x + 7
Example
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Q.1 Find slope, x intercept, y intercept of lines(ii) 2x + 7y – 3 = 0
Example
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Q.1 Find slope, x intercept, y intercept of lines(iii) line joining point (1,1), (9, 3)
Example
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Double Intercept Form
![Page 225: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/225.jpg)
Double Intercept Form
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Q.1 Find equation of straight line through (1, 2)& if its x intercept is twice the y intercept.
Example
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Q.2 Find equation of line passing through (2,3)and having intercept of y axis twice itsintercept on x axis
![Page 228: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/228.jpg)
Normal Form
![Page 229: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/229.jpg)
x cosα + ysinα = p ; α ∈ [0, 2π)
Normal Form
![Page 230: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/230.jpg)
Q.1 If equation of line is 3x – 4y + 5 = 0convert line in(i) Intercept form
Example
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Q.1 If equation of line is 3x – 4y + 5 = 0convert line in(ii) Double intercept form
Example
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Q.1 If equation of line is 3x – 4y + 5 = 0convert line in(iii) Normal form
Example
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Note
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(1) Line having equal intercept⇒ m = -1
Note
![Page 235: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/235.jpg)
(1) Line having equal intercept⇒ m = -1(2) Line equally inclined with coordinate axis
⇒ m = ± 1
Note
![Page 236: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/236.jpg)
Q.1 Find equation medians of∆ABC wherecoordinate of vertices are (0,0), (6,0), (3,8)
Example
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Q.2 If p is perpendicular distance fromoriginupon line whose intercept on coordinate axisare a & b prove that
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Q.3 Find locus of middle point of ABwhere A isx intercept and B is Y intercept of a variableline always passing through point (2,3)
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Q.4 Find number of lines passing through (2,4) &forming a triangle of area 16 units with thecoordinate axis.
![Page 240: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/240.jpg)
Q.5 Find equation of line(i) Cuts off intercept 4 on x axis and
passing through (2, -3)
![Page 241: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/241.jpg)
Q.5 Find equation of line(ii) Cuts off equal intercept on coordinate
axis and passes through (2, 5)
![Page 242: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/242.jpg)
Q.5 Find equation of line(iii) Makes an angle 135° with positive x
axis and cuts y axis at a distance 8from the origin
![Page 243: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/243.jpg)
Q.5 Find equation of line(iv) Passing through (4,1) and making with
the axes in the first quadrant a trianglewhose area is 8
![Page 244: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/244.jpg)
Q.6 Find equation of the two lines which joinorigin and points of trisection of the portionof line x + 3y – 12 = 0 intercepted betweenco-ordinate axis.
![Page 245: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/245.jpg)
Line inDeterminant FormDeterminant Form
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Equation of Median ThroughA (x
1, y
1) in ∆∆∆∆ ABC
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Equation of Median ThroughA (x
1, y
1) in ∆∆∆∆ ABC
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Equation of Internal Angle Bisector
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Equation of Internal Angle Bisector
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Equation of External Angle Bisector
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Equation of External Angle Bisector
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Equation of the Altitude
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Equation of the Altitude
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Assignment - 4Assignment - 4
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Find the equation to the straight line
Q.1 Cutting off an intercept unity fromthepositive direction of the axis of y and inclinedat 45° to the axis of x.
Q.2 Cutting off an intercept-5 from the axis of yQ.2 Cutting off an intercept-5 from the axis of yand being equally inclined to the axes.
Q.3 Cutting of an intercept 2 fromthe negativedirection of the axis of y and inclined at 30°to OX.
![Page 256: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/256.jpg)
Q.4 Cutting off an intercept -3 fromthe axis of y
and inclined at an angle to the axis of x.
Find the equation to the straight lineQ.5 Cutting off intercepts 3 and 2 fromthe axes.Q.6 Cuttingoff intercepts-5 and6 from theaxes.Q.6 Cuttingoff intercepts-5 and6 from theaxes.Q.7 Find the equation to the straight line which
passes through the points (5, 6) and hasintercepts on the axes.(i) equal in magnitude and both positive.(ii) equal in magnitude but opposite in sign.
![Page 257: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/257.jpg)
Q.8 Find the equations to the straight lines which
passes through the point (1, -2) and cut off
equal distance fromthe two axes.
Q.9 Find the equation to the straight line whichQ.9 Find the equation to the straight line which
passes through the given point (x′, y′) and is
such that the given point bisects the part
intercepted between the axes.
![Page 258: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/258.jpg)
Q.10 Find the equation to the straight line which
passes through the point (-4, 3) and is such
that the portion of it between the axes is
dividedby thepointof theratio5 : 3.dividedby thepointof theratio5 : 3.
Trace the straight line whose equation are
Q.11 x + 2y + 3 = 0 Q.12 5x – 7y – 9 = 0
Q.13 3x + 7y = 0 Q.14 2x – 3y + 4 = 0
![Page 259: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/259.jpg)
Find the equations to the straight lines passing
through the following pairs of points
Q.15 (0, 0) and (2, -2)
Q.16 (3, 4) and (5, 6)
Q.17 (-1, 3) and (6, -7)
Q.18 (0, -a) and (b, 0)
Q.19 (a, b) and (a + b, a – b)
![Page 260: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/260.jpg)
Find the equation to the sides of the triangles the
coordinate of whose angular points are respectively.
Q.20 (1, 4), (2, -3), and (-1, -2)
Q.21 (0, 1), (2, 0), and (-1, -2)
Q.22 Find the equation to the diagonals of the
rectangle the equations of whose sides are
x = a, x = a′, y = b, and y = b′
![Page 261: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/261.jpg)
Q.23 Find the equation to the straight line which
bisects the distance between the points (a, b)
and (a′, b′) and also bisects the distance
between the points (-a, b) and (a′, -b′).
Q.24 Find the equations to the straight lines which
go through the origin and trisect the portion of
the straight line 3x+y = 12 which is
intercepted between the axes of coordinates.
![Page 262: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/262.jpg)
Angle Between Two Lines
![Page 263: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/263.jpg)
Angle Between Two Lines
![Page 264: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/264.jpg)
Condition of lines being ||
![Page 265: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/265.jpg)
Condition of lines being ||
m = mm1
= m2
![Page 266: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/266.jpg)
Condition of lines being Perpendicular
![Page 267: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/267.jpg)
Condition of lines being Perpendicular
m m = -1m1
m2
= -1
![Page 268: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/268.jpg)
Equation of line parallel toax + by + c = 0
![Page 269: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/269.jpg)
Equation of line parallel toax + by + c = 0
ax + by + λλλλ = 0ax + by + λλλλ = 0
![Page 270: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/270.jpg)
Angle of line Perpendicular toax + by + c = 0
![Page 271: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/271.jpg)
Angle of line Perpendicular toax + by + c = 0
bx – ay + λλλλ = 0bx – ay + λλλλ = 0
![Page 272: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/272.jpg)
Inclination of lines are complementary
![Page 273: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/273.jpg)
Inclination of lines are complementary
m m = 1m1
m2
= 1
![Page 274: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/274.jpg)
ExampleQ.1 Find equation of line parallel and perpendicular
to y = 3 and passing through (2, 7)
![Page 275: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/275.jpg)
Q.2 Find the equation of line parallel and⊥ to x = 1and passing through (-9, -3)
![Page 276: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/276.jpg)
Q.3 Find equation of line parallel and perpendicularto 2x + 3y = 7 and passing through (2, -3)
![Page 277: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/277.jpg)
Q.4 Line 2x+3y=7 intersects coordinate axis inA&B . Find perpendicular bisector of AB
![Page 278: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/278.jpg)
Q.5 A(0, 8), B(2, 4) & C(6,8) find equation ofaltitudes, ⊥ bisectors and Coordinates ofOrthocenter and Circumcenter
![Page 279: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/279.jpg)
To Find Tangent of Interior Angles of Triangle
![Page 280: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/280.jpg)
To Find Tangent of Interior Angles of Triangle
![Page 281: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/281.jpg)
To Find Tangent of Interior Angles of Triangle
![Page 282: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/282.jpg)
To Find Tangent of Interior Angles of Triangle
![Page 283: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/283.jpg)
ExampleQ.1 If a∆ ABC is formed by the lines
2x + y – 3 = 0, x – y + 5 = 0 and 3x – y + 1 = 0then obtain tangents of the interior angles of thetriangle
![Page 284: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/284.jpg)
Q.2 Equation of line passing through (1, 2) makingan angle of 450 with the line 2x + 3y = 10
![Page 285: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/285.jpg)
Assignment - 5Assignment - 5
![Page 286: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/286.jpg)
Find the angles between the pairs of straight lines :
Q.1
Q.2 x – 4y = 4 and 6x – y 11
Q.3 y = 3x + 7 and 3y – x = 8
Q.4
Q.5 (m2 – mn)y = (mn + n2)x + n3 and (mn + m2)
= (mn – n2)x + m3
![Page 287: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/287.jpg)
Q.6 Find the tangent of the angle between the lines
whose intercepts on the axes are respectively
a, -b and b, -a.
Q.7 Prove that the points (2, -1), (0, 2), (2, 3) and
(4, 0) are the coordinates of the angular points
of a parallelogramand find the angle between
its diagonals.
![Page 288: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/288.jpg)
Find the equation to the straight line
Q.8 Passing through the point (2, 3) and
perpendicular to the straight line 4x – 3y = 10
Q.9 Passing through the point (-6, 10) and
perpendicular to the straight lines 7x + 8y = 5.
Q.10 Passing through the point (2, -3) and
perpendicular to the straight line joining the
points (5, 7) and (-6, 3)
![Page 289: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/289.jpg)
Q.11 Passing through the point (-4, -3) andperpendicular to the straight lines joining (1, 3)and (2, 7)
Q.12 Find the equation to the straight lines drawn atright anglesto thestraightline throughright anglesto thestraightline throughthe point where it meets the axis of x.
Q.13 Find the equation to the straight line whichbisects, and is perpendicular to the straight linejoining the points (a, b) and (a′, b′).
![Page 290: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/290.jpg)
Q.14 Prove that the equation to the straight line which
passes through the point (a cos3θ, a sin3θ) and is
perpendicular to the straight line x secθ + y
cosecθ = a is x cosθ - y sin θ = a cos 2θ.
Q.15 Find the equationsto the straight lines whichQ.15 Find the equationsto the straight lines which
divide, internally and externally, the line joining
(-3, 7) to (5, -4) in the ratio 4 : 7 and which are
perpendicular to this line.
![Page 291: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/291.jpg)
Q.16 Through the point (3, 4) are drawn two
straight lines each inclined at 45° to the
straight line x – y = 2. Find their equations
and find also the area included by the three
lines.lines.
Q.17 Showthat the equation to the straight line
passing through the point (3, -2) and inclined
at 60° to the line
![Page 292: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/292.jpg)
Q.18 Find the equations to the straight lines which
pass through the origin and are inclined at 75°
to the straight line
Q.19 Find the equations to the straight lines which
passthroughthepoint (h, k) andareinclinedatpassthroughthepoint (h, k) andareinclinedat
an angle tan-1m to the straight line y = mx + c.
![Page 293: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/293.jpg)
Reflection of a Point about a line
![Page 294: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/294.jpg)
Reflection of a Point about a line
![Page 295: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/295.jpg)
Reflection of a Point about a line
i. A B = B C
![Page 296: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/296.jpg)
Reflection of a Point about a line
i. A B = B C
ii. Angle 90o
![Page 297: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/297.jpg)
ExampleQ.1 Find equation of line passing through (-2, -7)
making an angle of with the line4x + 3y = 3
![Page 298: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/298.jpg)
Q.2 Find reflection of point (1, -2) about the linex – y + 5 = 0
![Page 299: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/299.jpg)
Q.3 Find reflection of point (1, -2) about the linex + 2y = 0
![Page 300: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/300.jpg)
Length of ⊥⊥⊥⊥ from (x1, y
1) on
ax + by + c = 0
![Page 301: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/301.jpg)
Length of ⊥⊥⊥⊥ from (x1, y
1) on
ax + by + c = 0
![Page 302: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/302.jpg)
Length of ⊥⊥⊥⊥ from (x1, y
1) on
ax + by + c = 0
![Page 303: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/303.jpg)
Distance BetweenTwo Parallel Lines
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Distance BetweenTwo Parallel Lines
![Page 305: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/305.jpg)
Distance BetweenTwo Parallel Lines
![Page 306: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/306.jpg)
Example1. Find distance between point (1, 2) and line
3x – 4y + 1 = 0
![Page 307: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/307.jpg)
2. Find distance between point (0, 0) and line 12x – 5y + 7 = 0
![Page 308: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/308.jpg)
3. Find distance between line 3x + 4y + 7 = 0 & 6x + 8y – 17 = 0
![Page 309: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/309.jpg)
Area of Parallelogram
![Page 310: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/310.jpg)
Area of Parallelogram
![Page 311: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/311.jpg)
Area of Parallelogram
![Page 312: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/312.jpg)
Note
Two parallel lines are tangent to same circle.Distance between them is diameter of the circle
![Page 313: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/313.jpg)
Note
Two parallel lines are tangent to same circle.Distance between them is diameter of the circle
![Page 314: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/314.jpg)
Note
Equation of diameter parallel to tangent
![Page 315: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/315.jpg)
Note
Equation of diameter parallel to tangent
![Page 316: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/316.jpg)
Note
Equation of diameter parallel to tangent
![Page 317: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/317.jpg)
Area of right isosceles ∆∆∆∆ interm of ⊥⊥⊥⊥ from vertex
![Page 318: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/318.jpg)
Area of right isosceles ∆∆∆∆ interm of ⊥⊥⊥⊥ from vertex
![Page 319: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/319.jpg)
Area of right isosceles ∆∆∆∆ interm of ⊥⊥⊥⊥ from vertex
∆∆ = p2
![Page 320: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/320.jpg)
Area of equilateral ∆∆∆∆ in terms of median / angle bisector /
⊥⊥⊥⊥ bisector / altitude
![Page 321: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/321.jpg)
Area of equilateral ∆∆∆∆ in terms of median / angle bisector /
⊥⊥⊥⊥ bisector / altitude
![Page 322: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/322.jpg)
Area of equilateral ∆∆∆∆ in terms of median / angle bisector /
⊥⊥⊥⊥ bisector / altitude
![Page 323: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/323.jpg)
ExampleQ.1 Find area of equilateral∆ whose one vertex is
(7, 0) & a side lies along line y = x
![Page 324: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/324.jpg)
Q.2 Two mutually ⊥ lines are drawn passingthrough points (a, b) and enclosed in anisosceles∆ together with the line x cosα + ysin α = p , Find the area of∆
![Page 325: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/325.jpg)
Q.3 The three lines x + 2y + 3 = 0, x + 2y – 7= 0and 2x – y – 4 = 0 formthe three sides of asquare, Find the equation of the fourth side
![Page 326: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/326.jpg)
Q.4 Find area of quadrilateral formed by the lines3x – 4y + 10 = 0, 3x – 4y + 20 = 0,4x + 3y + 10 = 0, 4x + 3y + 20 = 0
![Page 327: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/327.jpg)
Q.5 Find area of quadrilateral formed by the lines3x – 4y + 1 = 0, 3x – 4y + 2 = 0,x – 2y + 3 = 0, x – 2y + 7 = 0
![Page 328: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/328.jpg)
Parametric Form of Line / Distance Form
![Page 329: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/329.jpg)
Parametric Form of Line / Distance Form
![Page 330: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/330.jpg)
Parametric Form of Line / Distance Form
or x = x1 + r cos θ , y = y1 + r sin θ
![Page 331: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/331.jpg)
ExampleQ.1 In what direction a line through point (1, 2)
must be drawn so that its intersection point Pwith the line x + y = 4 may be at a distance of
fromA
![Page 332: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/332.jpg)
Q.2 If A(3,2), B(7,4), Find coordinate of C suchthat∆ ABC is equilateral.
![Page 333: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/333.jpg)
Q.3 A line passing through A (-5, -4) meets the linex + 3y + 2 = 0, 2x + y + 4 = 0 andx – y – 5 = 0 at B,C,D
If , find the equation of line
[IIT-JEE 1993]
![Page 334: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/334.jpg)
Q.4 Two side of a rhombus lying in 1st quadrant
are given by & . If the length
of longer diagonal OC= 12, Find the equation
of other two sides
![Page 335: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/335.jpg)
Assignments - 6
![Page 336: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/336.jpg)
Find the length of the perpendicular drawn from
Q.1 The point (4, 5) upon the straight line 3x + 4y = 10
Q.2 The origin upon the straight line
Q.3 The point (-3, -4) upon the straight line
12(x + 6) = 5(y – 2)12(x + 6) = 5(y – 2)
Q.4 The point (b, a) upon the straight line
Q.5 Find the length of the perpendicular fromtheorigin upon the straight line joining the twopoints whose coordinates are(a cosα, a sinα) and (a cosβ, a sinβ)
![Page 337: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/337.jpg)
Q.6 Shewthat the product of the perpendiculars
drawn fromthe two point
upon the straight line
Q.7 If p and p′ be the perpendicular fromthe origin
uponthestraightlineswhoseequationsareuponthestraightlineswhoseequationsare
x secθ + y cosecθ = a and
x cosθ - y sin θ = a cos2θ
Prove that : 4p2 + p′2 = a2
![Page 338: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/338.jpg)
Q.8 Find the distance between the two parallel
straight line y = mx + c and y = mx + d
Q.9 What are the point on the axis of x whose
perpendicular distance fromthe straight line
is a?is a?
Q.10 Find the perpendicular distance fromthe origin
of the perpendicular fromthe point (1, 2) upon
the straight line
![Page 339: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/339.jpg)
Position of point w.r.t. a Line
![Page 340: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/340.jpg)
Position of point w.r.t. a Line
![Page 341: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/341.jpg)
Position of point w.r.t. a Line
(ax2+by2+c) (ax3+by3+c) > 0
![Page 342: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/342.jpg)
Position of point w.r.t. a Line
(ax2+by2+c) (ax3+by3+c) > 0
(ax1+by1+c) (ax2+by2+c) < 0
![Page 343: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/343.jpg)
ExampleQ.1 Find range of a for which (a, a2) and origin lie
on same side of line 4x + 4y – 3 = 0
![Page 344: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/344.jpg)
Q.2 If point (a, a2) lies between lines
x + y – 2 = 0 & 4x + 4y – 3 = 0,
Find the range of a.
![Page 345: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/345.jpg)
Q.3 Determine values ofα for which point(α, α2) lies inside the triangle formed bythe lines 2x + 3y – 1 = 0, x + 2y – 3 = 0& 5x – 6y – 1 = 0 [IIT-JEE1992]
![Page 346: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/346.jpg)
Condition of Concurrency
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Condition of Concurrency
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Condition of Concurrency
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ExampleQ.1 Find k if lines x-y=3, x + y = 7, kx + 3y = 4
are concurrent
![Page 350: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/350.jpg)
Q.2 Prove that in any∆ altitudes are concurrent[IIT-JEE 1998]
![Page 351: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/351.jpg)
Q.3 Letλ, α∈R the lines λx + sinα y + cosα = 0x + cosα y + sinα = 0– x + sinα y – cosα = 0
If these lines are concurrent find the range ofλIf λ = 1 find general solution forα
![Page 352: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/352.jpg)
Q.4 If bc≠ ad and the lines (sin 3θ)x+ay+b=0(cos 2θ)x+cy+d=02x+(a+2c)y+(b+2d)=0
are concurrent then find the most generalvalues ofθ
![Page 353: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/353.jpg)
Family of lines
![Page 354: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/354.jpg)
Family of lines(i) Family of concurrent lines
![Page 355: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/355.jpg)
Family of lines(ii) Family of parallel lines
![Page 356: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/356.jpg)
ExampleQ.1 Find the point through which the line
x – 1 +λ y = 0 always passes through∀λ∈R
![Page 357: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/357.jpg)
Q.2 Find the point through which the linex–2+λ(y–1) = 0 always passes through∀λ∈R
![Page 358: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/358.jpg)
Q.3 Find the point through which the line2x – 3λ = y+7 always passes through∀λ∈R
![Page 359: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/359.jpg)
Type - 1Equation of line always passing through point ofintersection ofl
1= 0 & l
2= 0
is l1
+ λl2
= 0 ∀λ∈R
![Page 360: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/360.jpg)
ExampleQ.1 Find equation of line passing through
intersection of 3x – 4y + 6 = 0 & x + y + 2 = 0and(i) Parallel to line y = 0
![Page 361: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/361.jpg)
ExampleQ.1 Find equation of line passing through
intersection of 3x – 4y + 6 = 0 & x + y + 2 = 0and(ii) Parallel to line x = 7
![Page 362: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/362.jpg)
ExampleQ.1 Find equation of line passing through
intersection of 3x – 4y + 6 = 0 & x + y + 2 = 0and(iii) At a distance of 5 units fromorigin
![Page 363: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/363.jpg)
ExampleQ.1 Find equation of line passing through
intersection of 3x – 4y + 6 = 0 & x + y + 2 = 0and(iv) Situated at maximumdistance frompoint
(2,3)(2,3)
![Page 364: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/364.jpg)
Type – 2(Converse of Type - 1)
![Page 365: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/365.jpg)
Type – 2(Converse of Type - 1)
l1
+ λl2
= 0 always passes through intersection ofl1
= 0 & l2
= 0
![Page 366: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/366.jpg)
ExampleQ.1 If a, b, c are in A.P. Find the point through
which ax+by+c = 0 always passes through.
![Page 367: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/367.jpg)
Q.2 If a+3b = 5c, find the fixed point throughwhich line ax+by+c=0 passes
![Page 368: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/368.jpg)
Q.3 If a2 + 9b2 = 6ab + 4c2 then ax + by + c = 0passes through one or the other of which twofixed points ?
![Page 369: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/369.jpg)
Q.4 If algebraic sumof the perpendiculars fromA(x
1, y
1), B(x
2, y
2), C(x
3, y
3) on a variable
line ax + by + c = 0 vanishes then thevariable line always passes through.(A) G of ∆ABC (B) O of ∆ABC(C) I of ∆ABC (D) H of ∆ABC
![Page 370: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/370.jpg)
Q.5 The family of lines x(a+2b)+y(a+3b) = a+balways passes through a fixed point. Find thepoint.
![Page 371: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/371.jpg)
Q.6 The equations to the sides of a triangle arex + 2y = 0, 4x + 3y = 5 and 3x + y = 0.Find the coordinates of the orthocentre ofthe triangle without finding vertices oftriangle.
![Page 372: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/372.jpg)
Type - 3
![Page 373: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/373.jpg)
Type - 3
Equation of diagonals of ||gm
AC = µ1µ
2- µ
3µ
4= 0
![Page 374: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/374.jpg)
Type - 3
Equation of diagonals of ||gm
AC = µ1µ
2- µ
3µ
4= 0
BD = µ1µ
4- µ
2µ
3= 0
![Page 375: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/375.jpg)
ExampleQ. Find the equations of the diagonals of the
parallelogramformed by the lines2x – y + 7 = 0, 2x – y – 5 = 0, 3x + 2y – 5 = 0& 3x + 2y + 4 = 0
![Page 376: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/376.jpg)
Optics Problems
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ExampleQ.1 (i) Find reflection of point A (1,7) about y axes
![Page 378: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/378.jpg)
ExampleQ.1 (ii) Find reflection of point (10,3) about x axes
![Page 379: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/379.jpg)
ExampleQ.1 (iii) If A (1, 7) B (10, 3). Find coordinate of
point C & D
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Q.2 If A (1, 2) & B (3, 5), point P lies on x axisfind P such that AP+ PBis minimum
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Shifting of Origin
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Shifting of Origin
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Shifting of Origin
X = x – a
Y = y - b
![Page 384: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/384.jpg)
ExampleQ.1 Find the newcoordinate of point (3, -4) if
origin is shifted to (1,2)
![Page 385: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/385.jpg)
Q.2 Find transformed equation of the straight line2x-3y+5 = 0 if origin is shifted to (3, -1)
![Page 386: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/386.jpg)
Q.3 Find the point to which the origin should beshifted so that the equationx2 + y2 – 5x + 2y – 5 = 0 has no one degreeterms
![Page 387: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/387.jpg)
Q.4 Find the point to which the origin should beshifted so that the equationy2-6y-4x+13 = 0 is transformed to y2 = AX
![Page 388: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/388.jpg)
Q.5 Find area of triangle formed with vertices(2,0), (0,0), (1,4) if origin is shifted to (2010,2012)
![Page 389: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/389.jpg)
NoteSlope of line remains same after changing theorigin
![Page 390: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/390.jpg)
ExampleQ.1 If the axes are shifted to (1,1) then what do
the following equation becomes(i) x2 + xy – 3y2 – y + 2 = 0
![Page 391: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/391.jpg)
ExampleQ.1 If the axes are shifted to (1,1) then what do
the following equation becomes(ii) xy – x – y + 1 = 0
![Page 392: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/392.jpg)
ExampleQ.1 If the axes are shifted to (1,1) then what do
the following equation becomes(iii) x 2 – y2 – 2x + 2y = 0
![Page 393: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/393.jpg)
Rotation of Co-ordinate System
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Rotation of Co-ordinate System
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Rotation of Co-ordinate System
![Page 396: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/396.jpg)
ExampleQ. If the axes are rotated through an angle of
30° in the anticlockwise direction about theorigin. The co-ordinates of point arein the in new system. Find its oldcoordinates.coordinates.
![Page 397: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/397.jpg)
Angle Bisector
![Page 398: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/398.jpg)
Angle BisectorLocus of point such that its distance fromtwointersecting lines is equal
![Page 399: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/399.jpg)
Angle BisectorLocus of point such that its distance fromtwointersecting lines is equal
![Page 400: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/400.jpg)
ExampleQ. Find equation of angle bisector for lines
3x + 4y + 1 = 0, 12x + 5y + 3 = 0
![Page 401: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/401.jpg)
To discriminate between the acute & obtuse angle bisector
(Method – 2)
![Page 402: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/402.jpg)
To discriminate between the acute & obtuse angle bisector
(Method – 3)
![Page 403: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/403.jpg)
ExampleQ. Find the bisectors between the line
4x + 3y – 7 = 0 and 24x+7y-31=0.Identify acute/obtuse and origin containing/notcontaining
![Page 404: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/404.jpg)
To discriminate between the bisector of angle containing
origin and that of the angle not containing origin.
![Page 405: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/405.jpg)
To discriminate between the bisector of angle containing
origin and that of the angle not containing origin.
(i) Rewrite lines with same sign of absolute term.
![Page 406: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/406.jpg)
To discriminate between the bisector of angle containing
origin and that of the angle not containing origin.
(ii) Now positive sign will give you origincontaining angle bisector
![Page 407: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/407.jpg)
ExampleQ.1 The vertices of a∆ABC are
A(-1, 11), B(-9, -8) and C(15, -2)find the equation of the bisector of the angleat vertex A.
![Page 408: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/408.jpg)
Q.2 Find bisectors between the lines
and
![Page 409: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/409.jpg)
NoteIf m
1+ m
2= 0 ⇒ lines equally inclined with the
axes.
![Page 410: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/410.jpg)
Pair of Straight Line
![Page 411: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/411.jpg)
Pair of Line Through Origin
![Page 412: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/412.jpg)
Pair of Line Through Origin
ax2 + 2h xy + by2 = 0 (2° equation)
(i) h2 > ab⇒ lines are real & distinct
![Page 413: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/413.jpg)
Pair of Line Through Origin
ax2 + 2h xy + by2 = 0 (2° equation)
(ii) h2 = ab⇒ lines are coincidental
![Page 414: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/414.jpg)
Pair of Line Through Origin
ax2 + 2h xy + by2 = 0 (2° equation)
(iii) h 2 < ab⇒ lines are imaginary with real point
of intersectionof intersection
![Page 415: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/415.jpg)
Note
A homogeneous equation of degreen representnstraight lines passing through origin.
![Page 416: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/416.jpg)
NoteIf y = m
1x & y=m2x be two equation represented
by ax2+ 2h xy + by2 = 0 then
m1
+ m2
=
m1
m2
=
![Page 417: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/417.jpg)
Angle between two linesax2 + 2hxy + by2 = 0
![Page 418: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/418.jpg)
Angle between two linesax2 + 2hxy + by2 = 0
![Page 419: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/419.jpg)
Lines being perpendicular
![Page 420: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/420.jpg)
Lines being perpendicular
coefficient of x2 + coefficient of y2 = 0i.e. a + b = 0
![Page 421: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/421.jpg)
Lines are || / Coincident
![Page 422: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/422.jpg)
Lines are || / Coincident
h2 = ab
![Page 423: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/423.jpg)
Lines are equally inclined toX axis or coordinate axes
are bisectors
![Page 424: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/424.jpg)
Lines are equally inclined toX axis or coordinate axes
are bisectors
Coefficient of xy = 0Coefficient of xy = 0
![Page 425: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/425.jpg)
ExamplesQ.1 Find angle between lines given by
x2 + 4xy + 4y2 = 0
![Page 426: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/426.jpg)
Q.2 Find angle between lines given byx2 + 4xy + y2 = 0
![Page 427: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/427.jpg)
Q.3 Find angle between lines given by y2 - 3x2 = 0
![Page 428: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/428.jpg)
Q.4 Find angle between lines given by xy = 0
![Page 429: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/429.jpg)
Q.5 Find angle between lines given by3x2 + 10xy - 3y2 = 0
![Page 430: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/430.jpg)
Assignments - 7
![Page 431: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/431.jpg)
Find what straight lines are represented by the
following equations and determine the angles
between them.
Q.1 x2 – 7xy + 12y2 = 0
Q.2 4x2 – 24xy + 11y2 = 0Q.2 4x2 – 24xy + 11y2 = 0
Q.3 33x2 – 71xy – 14y2 = 0
Q.4 x3 – 6x2 + 11x – 6 = 0
Q.5 y2 – 16 = 0
Q.6 y3 – xy22 – 14x2y + 24x3 = 0
![Page 432: Straight Lines Slides-239](https://reader034.fdocuments.in/reader034/viewer/2022050706/545a33aeaf79592b448b5ba5/html5/thumbnails/432.jpg)
Q.7 x2 + 2xy secθ + y2 = 0
Q.8 x2 + 2xy cotθ + y2 = 0
Q.9 Find the equations of the straight lines bisecting
the angles between the pairs of straight lines
givenin example2, 3, 7 and8.givenin example2, 3, 7 and8.
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General Equation of 2°
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General Equation of 2°
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
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Condition that 2° equation represents pair of lines
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Condition that 2° equation represents pair of lines
abc + 2fgh – af2 – bg2 – ch2 = 0
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Condition that 2° equation represents pair of lines
abc + 2fgh – af2 – bg2 – ch2 = 0
or
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ExampleQ.1 Find whether 2x2 – xy – y2 – x + 4y – 3 = 0
can be factorized in two linears. If yes find thefactors. Also find the angle between the lines.
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Q.2 12x2 + 7xy – 10y2 + 13x + 45y – 35 = 0factorize this in two linears.
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Q.3 Find condition for whichax2 + bx2y + cxy2 + dy3 = 0represent three lines two of which are at rightangle.
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Q.4 Prove that 3x2 – 8xy – 3y2 = 0 and x + 2y = 3enclose a right isosceles∆. Also find area of∆.
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Q.5 Prove that the linesx2 – 4xy + y2 = 0 and x + y = 1 enclose anequilateral triangle. Find also its area.
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Q.6 Find centroid of∆ the equation of whose sidesare 12x2 – 20xy + 7y2 = 0 & 2x – 3y + 4 = 0
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Q.7 Find distance between parallel lines(i) 4x2 + 4xy + y2 – 6x – 3y – 4 = 0
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Q.7 Find distance between parallel lines(ii)
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Q.8 The equation ax2 + 6xy – 5y2 – 4x + 6y + c = 0represents two perpendicular straight lines find‘a’ and ‘c’.
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Equation of angles bisectors ofax2 + 2hxy + by2 = 0
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Equation of angles bisectors ofax2 + 2hxy + by2 = 0
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ExampleQ.1 Find equation of angle bisector of straight
lines xy = 0
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Q.2 Find equation of angle bisector of straightlines x2 – y2 = 0
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Product of ⊥⊥⊥⊥ dropped from(x
1, y
1) to line pair given by
ax2 + 2hxy + by2 = 0
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Product of ⊥⊥⊥⊥ dropped from(x
1, y
1) to line pair given by
ax2 + 2hxy + by2 = 0
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Homogenization
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Homogenization
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0lx + my + n = 0
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Homogenization
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0lx + my + n = 0
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ExampleQ.1 Find the equation of the line pair joining origin
and the point of intersection of the line 2x – y = 3and the curve x2 – y2 – xy + 3x – 6y + 18 = 0.Also find the angle between these two lines.
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Q.2 Find the value of ‘m’ if the lines joining theorigin to the points common tox2 + y2 + x – 2y – m = 0 & x + y = 1are at right angles.
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Q.3 Showthat all chords of the curve3x2 – y2 – 2x + 4y = 0subtending right angles at the origin passthrough a fixed point. Find also the coordinatesof the fixed point. [IIT-JEE1991]
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Q.4 A line L passing through the point (2, 1)intersects the curve 4x2 + y2 – x + 4y – 2 = 0 atthe points A, B. If the lines joining origin andthe points A, B are such that the coordinate axisare the bisectors between themthen find theequation of line L.
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Q.5 A straight line is drawn fromthe point (1,0) tointersect the curve x2 + y2 + 6x – 10y + 1 = 0such that the intercept made by it on the curvesubtend a right angle at the origin. Find theequation of the line L.
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Assignments - 8
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Prove that the following equations represent two
straight lines; find also their point of intersection and
the angle between them.
Q.1 6y2 – xy – x2 + 30y + 36 = 0
Q.2 x2 – 5xy + 4y2 + x + 2y – 2 = 0
Q.3 3y2 – 8xy – 3x2 – 29x + 3y – 18 = 0
Q.4 y2 + xy – 2x2 – 5x – y – 2 = 0
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Q.5 Prove that the equation,
x2 + 6xy + 9y2 + 4x + 12y – 5 = 0
represent two parallel lines.
Find the value of k so that the following equations
mayrepresentpairsof straightlines:mayrepresentpairsof straightlines:
Q.6 6x2 + 11xy – 10y2 + x + 31y + k = 0
Q.7 12x2 – 10xy + 2y22 + 11x – 5y + k = 0
Q.8 12x2 + kxy + 2y2 + 11x – 5y + 2 = 0
Q.9 6x2 + xy + ky2 – 11x + 43y – 35 = 0
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Q.9 kxy – 8x + 9y – 12 = 0
Q.10 x2 + xy + y2 – 5x – 7y + k = 0
Q.11 12x2 + xy – 6y2 – 29x + 8y + k = 0
Q.12 2x2 + xy – y2 + kx + 6y – 9 = 0
Q.13 x2 + kxy + y2 – 5x – 7y + 6 = 0Q.13 x2 + kxy + y2 – 5x – 7y + 6 = 0
Q.14 Prove that the equations to the straight lines
passing through the origin which make an angle
α with the straight lines y + x = 0 are given by
the equation, x2 + 2xy sec 2α + y2 = 0
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Q.15 The equations to a pair of opposite sides of a
parallelogramare :
x2 – 7x + 6 = 0 and y2 – 14y + 40 = 0
find the equations to its diagonals.