1. Circle Centre (a, b) radius r
Sunday, 02 December 2018

2. Equation of a circle through (a, b) with radius r

3. Example Find the equation the circle with centre (–2, 4) and radius 4.
𝑥+2 2
+ 𝑦−4 2
=16

4. Example Find the equation of the circle centre (–2, 4) which touches the y axis.
𝑥+2 2
+ 𝑦−4 2 =4
−2,4 2

5. Example
Show that the equation 𝑥 2 + 𝑦 2 −2𝑥+4𝑦−4=0
represents a circle and find its centre and radius.
𝑥 2 + 𝑦 2 −2𝑥+4𝑦−4=0
𝑥 2 −2𝑥 + 𝑦 2 +4𝑦 =4
𝑥 2 −2𝑥 𝑦 2 +4𝑦 =4+1+4
𝑥−1 2
+ 𝑦+2 2 =9
∴ Circle centre 1,−2 and radius 3

6. Example
Show that the equation 𝑥 2 + 𝑦 2 −16𝑥+10𝑦+85=0 represents a circle.
𝑥 2 + 𝑦 2 −16𝑥+10𝑦+85=0
𝑥 2 −16𝑥 + 𝑦 2 +10𝑦 =−85
𝑥 2 −16𝑥 𝑦 2 +10𝑦 =−
𝑥−8 2
+ 𝑦+5 2 =4
∴ Circle centre 8,−5 and radius 2

7. Example Find the points of intersection of the circle 𝑥 2 + 𝑦 2 =20 with the line 𝑦=3𝑥−2
Solve simultaneous equations
𝑥 2 + 𝑦 2 =20
𝑥 𝑥−2 2 =20
𝑥 2 + 9𝑥 2 −12𝑥+4=20
10𝑥 2 −12𝑥−16=0
5𝑥 2 −6𝑥−8=0
5𝑥+4
𝑥−2 =0
𝑥=− 4 5
𝑥=2
𝑦=4
𝑦=− 22 5
∴ points of intersection are 2,4 and −0.8,−4.4

8. Example
Work out the equation of the tangent that can be drawn to the circle with equation 𝑥 2 + 𝑦 2 +2𝑥−6𝑦−3=0 from the point 2,1
𝑥 2 + 𝑦 2 +2𝑥−6𝑦−3=0
𝑥 2 +2𝑥 + 𝑦 2 −6𝑦 =3
𝑥 2 +2𝑥 𝑦 2 −6𝑦 =3+1+9
𝑥+1 2
+ 𝑦−3 2
=13
∴ Circle centre −1,3 and radius 13

9. 1−−3 2−1 = 4 1 Gradient of radius = − 1 4 Gradient of tangent =
2,1
1,−3
Equation of tangent
− 1 4
𝑦− = 𝑥− 2 1
4𝑦−8 =−1 𝑥−1
4𝑦−8 =−𝑥+1
4𝑦+𝑥 =9

10. Questions
1. Find the centre and radius for the following circles
(a) 𝑥 2 + 𝑦 2 +2𝑥−8𝑦+13=0
(b) 𝑥 2 + 𝑦 2 −2𝑥−2𝑦−2=0

11. 2. Find the equations for each of the following circles
(a) centre 3,4 with radius 1
(b) centre −2,2 with radius 5

12. (c) centre 3,−2 with circle touching the x-axis
(d) Centre (1, 2) passing through the point (−1,5)

13. 3. Find the equation of the tangent to the circle
𝑥 2 + 𝑦 2 +6𝑥+3𝑦−4=0 at the point (−1,−2)

14. 4. Find the points of intersection of the circle 𝑥 2 + 𝑦 2 −4𝑥−1=0 and the line 𝑦=𝑥−5