Liceo Scientifico Isaac Newton
Maths course
The Circle
Professor
Serenella Iacino
Read by
Cinzia Cetraro
UTILIZZARE SPAZIO PER INSERIRE FOTO/IMMAGINE DI RIFERIMENTO LEZIONE
P(x;y)
Cr x² + y² + a ax + b by + c c = 0
aa, b b, c c
x
y
We define a circle as the geometric locus of the points P (x,y) equidistant from a point C, which is the center
1) a = b = 0 C
2) b = 0 C
C
3) a = 0
x² + y² + c = 0
x² + y² + ax + c = 0
x² + y² + by + c = 0
The different positions of a circle on a Cartesian Plane, according to the variation of its coefficients a, b, c
4) c = 0C
5) a = c = 0C
6) b = c = 0 C
x² + y² + ax + by = 0
x² + y² + by = 0
x² + y² + ax = 0
x = 2a y =
2b
r = ( )² + ( )² - c2)
P ( xp; yp )
r
x² + y² + a ax + b by + c c = 0
1)
3) P (xp; yp)
- -
2b-
2a-
x
y
To find the equation of a circle we need three independent conditions to determine the values of the constants, that is one condition for each constant
P1
C
P
CP2
P
C
x² + y² + ax + by + c = 0
y = mx + k
Δ = 0Δ > 0 Δ < 0
SecantSecant TangentTangent ExternalExternal
We can consider the position of a straight line in relation to a circle. It could be:
From an algebraic point of view we must solve this system:
P
RealReal
P
ImaginaryImaginary
P
CoincidentCoincident
x² + y² + a ax + b by + c c = 0
y - y = m m ( x - x ) P P
Now, we want to determine the equations of the two tangents to a circle from a given point P (xp ; yp )
From an algebraic point of view we must solve this system:
1) External
C C’
2) Tangent from the outside
C C’
Let’s consider now the position of two circles relative to each other which depending on the distance between their centres can be:
if the distance between their centres is greater than sum of the radii
if the distance between their centres is equal to the sum of the radii
3) Secant
C C’
4) Tangent within
CC’
5) Inside
C
C’
if the distance between their centres is less than the sum of the radii
if the distance between their centres is equal to the difference between the radii
if the distance between their centres is less than the difference between the radii
This is the system composed by the equations of the two circles
from which we obtain the equation of the radical axis with the elimination method:
x² + y² + a x + b y + c = 0 x² + y² + a’x + b’y + c’ = 0
x² + y² + a x + b y + c = 0 - x² - y² - a’ x - b’ y - c’ = 0
(a-a’)x+(b-b’)y+(c-c’)= 0(a-a’)x+(b-b’)y+(c-c’)= 0
CC’
CC’
CC’
SecantSecant TangentTangent ExternalExternal
radical axisradical axis
ξ’ ξ’ : x² + y² + a’x + b’y + c’ = 0
ξ ξ : x² + y² + a x + b y + c = 0
linear combination ξ ξ + λ ξ’ ξ’ = 0
First circle
From which, substituting ξ ξ and ξ’ ξ’ , we obtain:
Starting from these equations of two circles:
Second circle
x² + y² + a x + b y + c + ( x² + y² + a’x + b’y + c’ ) = 0λ
we can obtain through a linear combination linear combination the equation of a set of circles:set of circles:
( 1 + λ ) ( 1 + λ ) ( 1 + λ )= 0x² + y² + x + Y+
( a + λ a’ ) ( b + λ b’ ) ( c + λ c’ )
Grouping the terms of second degree, those of first degree and the constants, and dividing by (1+ (1+ λλ ) we obtain this equation:
which, for different values of λλ, represents all the circles passing through the intersections of ξξ and ξ’ ξ’ which are the Base Points of the set
except for λ = -1λ = -1
( a - a’ )x + ( b - b’ )y + ( c - c’ ) = 0 Radical AxisRadical Axis
In fact, for this value λ = -1λ = -1 we obtain the straight line:
Degenerate CircleDegenerate CircleCC C’C’
This Radical Axis can be considered as a particular circle having an infinite radius, and can be called Degenerate Circle
This lesson was prepared by prof.ssa Serenella Iacino
for the Liceo Scientifico Statale “Newton” of Rome..
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