WAEC 2010 · Paper 2 · Q9

Points $A(2, 1)$ and $B(4, -5)$ lie on a circle. If the line $2x - y - 13 = 0$ is a tangent to the circle at $B$ , find the:

(a)
coordinates of the centre of the circle;
Separate values with commas, e.g. 3, −2
(b)
equation of the circle.
Show the answer
$x^2 + y^2 + 6y - 11 = 0$

Worked solution (try it first)

Let the centre be $(x, y)$ .
The centre is the same distance from $A$ and $B$ : $(x - 2)^2 + (y - 1)^2 = (x - 4)^2 + (y + 5)^2$ .
Expand and simplify: $-4x - 2y + 5 = -8x + 10y + 41$ , so $x - 3y = 9$ .
The tangent $2x - y - 13 = 0$ has gradient 2, and the radius to $B$ is perpendicular to it, so the gradient of the radius is $-\frac12$ .
So $\dfrac{y + 5}{x - 4} = -\dfrac12$ , which gives $x + 2y = -6$ .
Subtract the two equations: $5y = -15$ , so $y = -3$ , and then $x = 9 + 3(-3) = 0$ .
The centre is $(0, -3)$ .