Coordinate Geometry Ver 2012

7/29/2019 Coordinate Geometry Ver 2012

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My

AdditionalMathemati

csModules

Form 4(Version 2012)

Topic 6:

Mastering

CoordinatesGeometry

by

NgKL(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH)



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COORDINATES GEOMETRY(GEOMETRI KOORDINAT)

IMPORTANT NOTES (1)


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Exercise 6.1 Distance Between Two Points

1. Find the distance between the following pairs of points.

(a) A (6, 4) and B (1, -8) (b) A (6, 8) and B (3, 4)

(c) C (9, -2) and D (3, 6) (d) P (7, -4) and Q (-5, 1)

2. Find the possible values of kfor each of the following situation.

(a) A (k, 4k), B (-k, 5k) and80=

AB unit (b) A (3, 1), B (6, k) and AB = 5 unit

(c) A (2, 3), B (k, 5) and 20=AB unit (d) A (-2, 1), B (2, k) and AB = 5 unit


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Exercise 6.2 Division of a Line Segment

1. Find the coordinate of the mid-point for straight lines of the following pairs of points:

(a) A (3, -2) and B (-7, -6) (b) P (-2, 7) and Q (4, -3)

(c) T (6, 4) and U (-2, 2) (d) V (-2, -3) and W (-6, -7)

2. Given that M is the mid point for the straight line AB. Find the coordinate of point B.

(a) A (-1, 2) dan M (-4, 5) (b) A (2, 3) dan M (5, 4)

(c) A (-2, 4) dan M (1, 5) (d) A (-6, 3) dan M (-2, -2)


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3. ABCD is a parallelogram. Find the values of nilai h and k.

(a) A(2, 3), B(5, 4), C(6, 7) and D(h, k) (b) A(-1,2), B(3, 5), C(h, k) dan D(-3, 7)

4. Find the coordinate of point P which divides the straight line AB to the ratio AP : PB as shown.

(a) A(3, 5), B(3, -2) and AP : PB = 2 : 5 (b) A(-2, -2), B(3, 3) and AP : PB = 4 : 1

(c) A(-1, 5), B(4, 5) and AP : PB = 2 : 3 (d) A(-6, -9), B(2, 3) and AP : PB = 5 : 3

5. Solve the following problems.


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(a) GivenA(m, 7), B(1, n) and line AB is divided by pointP(5, 3) according to the ratioAP : PB = 2 : 3. Find the values of m and n.

(b) GivenP(x, 5), Q(8, y) and linePQ is divided by pointM(1 ,7) according to the ratioPM : MQ = 1 : 3. Find the values ofx and y.

(c) GivenK( p, 4 ), L(3, n ) and lineKL is divided by pointM( 2, 6 ) according to the ratio

KM : 2ML. Find the values ofp and q.

Exercise 6.3 Areas of Polygons (Triangles and Quadrilaterals)


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1. Find the area of each of the following polygons with the given vertices.

(a) A(-2, 4), B(0, -6) C(8, 4) (b) A(7, 0), B(1, 2), C(2, -5)

(c) A(2, 5), B(6, 0), C(1, -3), D(-2, 2) (d) A(1, 6), B(1, 2), C(4, 1), D(6, 4)

2. Find the values of the unknown of a point with which the area of the triangle formed is given.

(a) K(-2, 3), L(5, 1) M(p, 5),Area of KLM = 18 units2.

(b) A(3, h), B(2, 7), C(9, 10),Area of ABC = 8.5 unit2.

3. Determine whether the following sets of points is collinear.


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(a) A(2, 1), B(14, 4) C(30, 8) (b) A(3, 5), B(2, 6), C(5, 4)

4. Find the value of the unknown of a point which formed the sets of collinear points

(a) A(5, w), B(1, 1) C(3, 3) (b) A(4, 0), B(0,y), C(2, 3)

(c) A(12, 5), B(r, 3), C(12, 1) (d) A(9, 6), B(3, s), C(4, 1), D(5, 4)

COORDINATE GEOMETRY(GEOMETRI KOORDINAT)

IMPORTANT NOTES (2)


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Exercise 6.4 Gradient and Equations of Straight Lines


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1. Find the gradient of the following pairs of points. Hence, find the equations of the straight lines.

(a) (2, 6) and (8, 9) (b) (-2, 2) and (6, -2)

(c) (-1, 3) and (-3, -1) (d) (3, -4) and (1, 0)

2. Diagrams show thex-interceptandy-interceptof each of the straight lines. Find each of the gradients byusing the intercepts, hence find the equation of the straight lines.

3. Given are pairs of the points which indicate thex-interceptand they-intercept. Form the equation of each ofthe straight lines in intercept form.

(a)y

5

0 4

(b) y

7 0

5

x

x


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15

y

3

x=+

17

y

5

x=+

14

y

2

x=


(a) A (3, 0) and B (0, 6) (b) P (2, 0) and Q (0, 3)

(c) T (6, 0) and U (0, 2) (d) V (2, 0) and W (0, 7)

4. Given are the equations of straight lines in their intercept form. State thex-intercept, y-interceptand find thegradient of each of the straight lines.

Equation in

intercept form x-intercept y-intercept gradient

a.

b.

c.

5. Find the equation of the straight line which has the gradient, mand passes through a point P as shown below.

(a) m = 4, P (-7, -6) (b) m = 3, P (4, -3)

(c) m = ,, P (6, 4) (d) m = 2, P (6, 7)

6. Given are the equations of straight lines in their gradient form. State the gradient,y-intercept,and x-interceptof each of the straight lines.


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14

y

2

x=


Equation inintercept form

x-intercept y-intercept gradient

a.y = 7x + 3

b. y = 3x + 5

7. Transform the given straight line equations to the respective forms as indicated in the table below.

Equation in intercept form Equation in gradient form Equation in general form

a. y = 3x + 7

b. 2x + 5y 4 = 0

c.

8. Pointof Intersection of a Pair of Straight Lines Solve by Simultaneous Equations Method.

Given are a pair of equations of straight lines. Find the coordinate when the straight lines intersect each other.

Exercise 6.5 Parallel and Perpendicular Lines

1. Determine whether each of the following pair of straight lines are parallel or perpendicular to each other.

(a) 2x + 3y = 21x + y = 8 (b) x 2y = 14x + y + 5 = 0


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63

y

2

x=+

16

y

3

x=+

16

y

3

x=+


(a) 2y + 3x = 12; (b) 2x + y =2; 4x 2y = 3

(c) 3y + x 5 = 0 ; y = 3x + 1 (d) ; 2y + 4x = 3

2. Solve the following problems.

(a) Find the equation of the straight line which passes through pointP(1, 3), and is parallel to3x 2y = 6.Write the equation in general form.

(b) Find the equation of the straight line which passes through pointP(4, 3), and is parallel to .Write the equation in gradient form.


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(c) Find the equation of the straight line which passes through pointP(2, 4), and is perpendicular to

3x + y = 6. Write the equation in general form.

(d) Find the equation of the straight line which passes through pointP(0, 6 ), and is perpendicular to

2x y +3 = 0. Write the equation in gradient form.

Exercise 6.6 Equation of Locus

1. Determine the equation of the locus of the moving pointP(x, y) that satisfies the following condition.

(a) PointP(x, y) moves suchthat it is 3 units frompoint Q (1, 5).

(b) PointP(x, y) moves such that it is 5 units frompointR(2, 5).

(c) PointP(x, y) moves such that its distance frompointA(4,8) is twice its distance from point B(0,3).

(d) PointP(x, y) moves such that it is equidistant frompointM(2, 4) andN(3, 12).


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(e) PointP(x, y) moves such that its distance frompointK(5,0) and pointL(1, 4) is in the ratio

PK : PL = 1 : 3.

(d) PointP(x, y) moves such that its distance frompointR(2, 4) is 6 units and point S(3, 12) is 4unit.

Exercise 6.7 Past Year SPM Questions

1. SPM2009/Paper 1/Q15y

Diagram 15 shows a straight line AC. A(2, 3)The point B lies on AC such that AB : BC = 3 : 1Find the coordinates of B.

B(h, k)

xO C(14, 0)

[3 marks]

Diagram 15

2. SPM2008/Paper 1/Q13.

Diagram 13 shows a straight line passing through S(3, 0) and T (0, 4).

(a) Write down the equation of the straight line y

in the form 1

b

y

a

x=+ . T(0, 4)

(b) A pointP(x, y) moves such that PS = PT. Find the equation of the locus of P.

x

O S(3, 0)

Diagram 13

[4 marks]


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Diagram 9 shows a trapezium OABC. The line OA is perpendicular to line AB, which intersects the y-axis at

point Q. It is given that equation of OA isy = -2

3x and the equation ofAB is 6y = kx + 26.

y(a) Find B (i) the value of k.

6y= kx + 26

Q

A

y = x2

3 C

x

(ii) the coordinate of A [4 marks]

(b) GivenAQ : QB = 1 : 2, find(i) the coordinate of B.

(ii) the equation of the straight line BC. [4

marks]

(c) A pointP(x, y) moves such that 2PA = PB. Find the equation of the locus ofP. [2

marks]

O


3. SPM2009/Paper 2/Q9

Solution by scale drawing is not accepted.

Diagram 9


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OO



Solution by scale drawing is not accepted. y P(2

3, k)

Diagram 10 shows a triangle OPQ. Point S lieson the line PQ.

(a) A point W moves such that its distance S(3, 1)from point S is always 2 units.Find the equation of the locus of W. [3 marks]

x

Q

(b) It is given that point P and point Q lie on the locus of W. Calculate(i) the value of k.

(ii) the coordinate of Q. [5 marks]

(c) Hence, find the area, in unit2, of triangle OPQ. [2

marks]

Diagram 10


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The straight line 1h

y

6

x=+ has a y-intercept of 2 and parallel to the straight line y = kx = 0. Determine the

value of h and of k. [3marks]


Solution by scale drawing will not be accepted.

In Diagram 1, the straight line AB has an equation y

y + 2x + 8 = 0 . AB intersects thex-axis at point Aintersects they-axis at point B.

Ax

PointP lies onAB such thatAP : PB = 1 : 3. P Find

(a) the coordinates of P. [3 marks]

By + 2x + 8 =

0

(b) the equation of the straight line that passes throughP

and is perpendicular toAB. [3 marks]

Diagram 1


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The straight line passes through A(2, 5) and B(6, 7).

(a) Given C (h,10) lies on the straight line AB. Find the value of h.

(b) Point D divides the lines segment AB in the ratio 1: 3,

Find the coordinates of D. [4marks]


Solution by scale drawing will not be accepted.

Diagram 5 shows the straight line AC which intersect the y-axis at point B.

The equation of AC is 3y = 2x 15 .

Find

(a) the equation of the straight line which passes through point A and is perpendicular to AC [4marks]

y

C

x

(b) (i) the coordinates of B.

A(3, 7)

(ii) the coordinates of C, given AB : BC = 2 : 7 [3 marks]

0

B

Diagram 5

Coordinate Geometry Ver 2012

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