Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — centre and radius from the general form
Find the centre and radius of the circle x2+y2−6x+4y−12=0.
- 1Complete the square in x: x2−6x=(x−3)2−9Half of −6 is −3.
- 2Complete the square in y: y2+4y=(y+2)2−4Half of 4 is 2.
- 3(x−3)2−9+(y+2)2−4−12=0
- 4(x−3)2+(y+2)2=25Collect the constants: −9−4−12=−25.
- 5Centre (3,−2), radius 5.The signs flip out of the brackets, and the radius is the square root of 25.
Example 2 — the tangent at a point on a circle
Find the tangent to x2+y2−6x+4y−12=0 at the point (7,1).
- 1Check the point lies on the circle: (7−3)2+(1+2)2=16+9=25 ✓Always confirm this first.
- 2The centre is (3,−2).From completing the square.
- 3Gradient of the radius: 7−31−(−2)=43
- 4A tangent is perpendicular to the radius, so its gradient is −34.Flip and change the sign.
- 5y−1=−34(x−7)
- 64x+3y=31Check at (7,1): 28+3=31 ✓
Example 3 — when a line is a tangent
Find the values of k for which y=x+k is a tangent to x2+y2=8.
- 1Substitute the line into the circle.
- 2x2+(x+k)2=8
- 32x2+2kx+k2−8=0Expand and collect.
- 4A tangent touches once, so the discriminant is zero.This is the key translation from geometry to algebra.
- 5(2k)2−4(2)(k2−8)=0
- 64k2−8k2+64=0, so k2=16 and k=±4.Two tangents, one on each side of the circle.