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

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

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

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

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+ 1 2 + 𝑦 2 +4𝑦+ 2 2 =4+1+4 𝑥−1 2 + 𝑦+2 2 =9 ∴ Circle centre 1,−2 and radius 3

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+ 8 2 + 𝑦 2 +10𝑦+ 5 2 =−85+64+25 𝑥−8 2 + 𝑦+5 2 =4 ∴ Circle centre 8,−5 and radius 2

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 2 + 3𝑥−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

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+ 1 2 + 𝑦 2 −6𝑦+ 3 2 =3+1+9 𝑥+1 2 + 𝑦−3 2 =13 ∴ Circle centre −1,3 and radius 13

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

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

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

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

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

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