Question

Find the centre of the circle passing through the points A(6, - 6), B(3, - 7) and X(3, 3).
The centre of the circle is assumed to be X(x,y)

Answer 1

Let A(6,-6), B(3,-7) and C(3.3) be the points on the circle and the centre of the circle be P (h,k).
$\therefore PA=PB=PC$ (Radius of the circle)
$\Rightarrow \sqrt{(h-6{)}^2+(k+6{)}^2}=\sqrt{(h-3{)}^2+(k+7{)}^2}=\sqrt{(h-3{)}^2+(k-3{)}^2}\Rightarrow (h-6{)}^2+(k+6{)}^2=(h-3{)}^2+(k+7{)}^2=(h-3{)}^2+(k-3{)}^2\Rightarrow 12h+12k+72=-6h+9+14k+49=-6h+9-6k+9\Rightarrow -12h+12k+72=-6h+14k+58=-6h-6k+18\therefore h+14k+58=-6h-6k+18\Rightarrow 20k=-40\Rightarrow k=-2$
Now, $-12h+12k+72=-6h-28+58\Rightarrow -6h=-18\therefore h=3$
Thus, the centre of the circle is (3,-2).