Notes Aidan Roche 2009 1 c Aidan Roche 2009 Given the centre and radius of a circle to find the equation of Circle K K r Method Sub centre amp radius into x h 2 y k ID: 551795
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Slide1
Co-ordinate Geometry of the CircleNotes
Aidan Roche2009
1
(c) Aidan Roche 2009Slide2
Given the centre and radius of a circle, to find the equation of Circle K?
K
r
Method
Sub centre & radius into:
(x – h)
2
+ (y – k)
2
= r
2 If required expand to: x2 + y2 +2gx +2fy + c = 0
c(h, k)
2
(c) Aidan Roche 2009Slide3
To find the centre and radius. Given the Circle K:
(x – h)2
+ (y – k)2 = r
2
Method
Centre: c(h, k)
Radius = r
K
r
c
3
(c) Aidan Roche 2009Slide4
To find the centre and radius. Given the Circle K:
x2
+ y 2
= r
2
Method
Centre: c(0, 0)
Radius = r
K
r
c
4(c) Aidan Roche 2009Slide5
To find centre and radius of K. G
iven the circle K:
x2 + y
2
+2gx +2fy + c = 0
?
K
Method
Centre: c(-g, -f)
Radius:
rc
5
(c) Aidan Roche 2009Slide6
Given equation of circle K, asked if a given point is on, inside or outside the circle?
a
Method
Sub each point into the circle formula K = 0
Answer > 0
outside
Answer = 0
on
Answer < 0 inside
b
c
K
6
(c) Aidan Roche 2009Slide7
Important to remember
Theorem
Angle at centre is twice the angle on the circle standing the same arc
c
7
(c) Aidan Roche 2009
θ
2
θ
a
b
dSlide8
Important to remember
Theorem
Angle on circle standing the diameter is 90
o
diameter
8
(c) Aidan Roche 2009
90
oSlide9
To find equation of circle K given end points of diameter?
K
Method
Centre is midpoint [
ab
]
Radius is ½|ab|
Sub into circle formula
a
9
(c) Aidan Roche 2009b
c
rSlide10
To prove a locus is a circle?
Method
If the locus of a set of points is a circle it can be written in the form:
x
2
+ y
2
+2gx + 2fy + c = 0
We then can write its centre and radius.
cK10(c) Aidan Roche 2009
rSlide11
To find the Cartesian equation of a circle given Trigonometric Parametric equations?
Method
Trigonometric equations of a circle are always in the form:
x = h ± rcosѲ
y = k ± rsinѲ
Sub h, k and r into Cartesian equation:
(x – h)
2
+ (y – k)
2
= r2cK11
(c) Aidan Roche 2009
rSlide12
To prove that given Trigonometric Parametric equations
(
x = h ± rcosѲ, y = k ± rsinѲ)
represent a circle?
Method
Rewrite cosѲ
(in terms of x, h & r)
and then evaluate cos
2Ѳ.
Rewrite sinѲ (in terms of y, h & r) and then evaluate sin2Ѳ.
Sub into: sin2Ѳ + cos2Ѳ = 1 If it’s a circle this can be written in the form: x2 + y2 +2gx + 2fy + c = 0
c
K
12
(c) Aidan Roche 2009
rSlide13
To find the Cartesian equation of circle
(in the form: x
2
+ y
2
= k)
given algebraic parametric equations?
Method
Evaluate: x2
+ y2The answer = r2
Centre = (0,0) & radius = rcK13
(c) Aidan Roche 2009
rSlide14
Given equations of Circle K and Circle C, to show that they touch internally?
K
Method
Find distance between centres
If
d = r
1
- r
2
QEDC
r
1
r
2
d
14
(c) Aidan Roche 2009
c
1
c
2Slide15
Given equations of Circle K and Circle C, to show that they touch externally?
K
Method
Find distance d between centres
If
d = r
1
+ r
2
QEDC
r
1
r
2
d
15
(c) Aidan Roche 2009
c
1
c
2Slide16
Given circle K and the line L to find points of intersection?
a
Method
Solve simultaneous equations
b
L
K
16
(c) Aidan Roche 2009Slide17
Important to remember
Theorem
A line from the centre (c) to the point of tangency (t) is perpendicular to the tangent.
c
17
(c) Aidan Roche 2009
90
o
Tangent
K
radius
tSlide18
Important to remember
Theorem
A line from the centre perpendicular to a chord bisects the chord.
c
18
(c) Aidan Roche 2009
90
o
a
b
radius
dSlide19
Given equation of Circle K and equation of Tangent T, find the point of intersection?
K
T
Method
Solve the simultaneous equations
19
(c) Aidan Roche 2009
tSlide20
Given equation of Circle K and asked to find equation of tangent T at given point t?
K
t
Method 1
Find slope [ct]
Find perpendicular slope of T
Solve equation of the line
c
T
Method 2
Use formula in log tables
20
(c) Aidan Roche 2009Slide21
To find equation of circle K, given that x-axis is tangent to K, and centre c(-f, -g) ?
X-axis
Method
On x-axis, y = 0 so t is (-f, 0)
r = |f|
Sub into circle formula
c(-g, -f)
K
21
(c) Aidan Roche 2009
t(-g, 0)
r = |f|Slide22
To find equation of circle K, given that y-axis is tangent to K, and centre c(-f, -g) ?
y-axis
Method
On y-axis, x = 0 so t is (0, -g)
r = |g|
Sub into circle formula
c(-g, -f)
K
22
(c) Aidan Roche 2009
t(0, -f)
r = |g|Slide23
Given equation of Circle K and equation of line L, how do you prove that L is a tangent?
K
L
Method 2
Find distance from c to L
If
d = r
it is a tangent
23
(c) Aidan Roche 2009
r
Method 1
Solve simultaneous equations and find that there is only one solution
cSlide24
Given equation of Circle K & Line L:
ax + by + c = 0
to find equation of tangents parallel to L?
K
r
Method 1
Find centre c and radius r
Let parallel tangents be:
ax + by + k = 0
Sub into distance from point to line formula and solve:
c
L
24
(c) Aidan Roche 2009
T
1
T
2
rSlide25
Given equation of Circle K and point p, to find distance d from a to point of tangency?
K
c
t
Method
Find r
Find |cp|
Use Pythagoras to find d
p
T
r
|cp|
d?
25
(c) Aidan Roche 2009Slide26
Given equation of Circle K and point p, to find equations of tangents from p(x
1,y1)?
K
c
p
T
1
r
26
(c) Aidan Roche 2009
T
2
r
Method 1
Find centre c and radius r
Sub p into line formula and write in form T=0 giving:
mx
– y + (mx
1
– y
1
) = 0
Use
distance from point to line formula
and solve for m:
Slide27
Given equation of Circle K and Circle C, to find the common Tangent T?
K
T
Method
Equation of T is:
K – C = 0
C
27
(c) Aidan Roche 2009Slide28
Given equation of Circle K and Circle C, to find the common chord L?
K
L
C
Method
Equation of T is:
K – C = 0
28
(c) Aidan Roche 2009Slide29
Given three points and asked to find the equation of the circle containing them?
a
Method
Sub each point into formula:
x
2
+ y
2
+ 2gx + 2fy + c = 0
Solve the 3 equations to find: g, f and c,
Sub into circle formulab
c
29
(c) Aidan Roche 2009Slide30
Given 2 points on circle and the line L containing the centre, to find the equation of the circle?
a
Method
Sub each point into the circle:
x
2
+ y
2
+ 2gx + 2fy + c = 0
Sub (-g, -f) into equation of L
Solve 3 equations to find: g, f and c, Sub solutions into circle equation
b
L
30
(c) Aidan Roche 2009Slide31
Given the equation of a tangent, the point of tangency and one other point on the circle, to find the equation of the circle?
a
Method
Sub each point into the circle:
x
2
+ y
2
+ 2gx + 2fy + c = 0
Use the tangent & tangent point to find the line L containing the centre.
Sub (-g, -f) into equation of LSolve 3 equations to find: g, f and c, Sub solutions into circle equation
b
T
31
(c) Aidan Roche 2009
L