Question

Find the equation of the circle which passes through (3, - 2), (-2, 0) and has its centre on the line 2x -y = 3

Answer 1

A circle passing through P(3, -2) and Q(-2, 0) and having its centre on 2x - y = 3.
Let the equation of the circle be
$x^2+y^2+2gx+2fy+c=0$
Since the circle passes through (3, -2) and also (-2, 0) therefore
$9+4+6g—4f+c=0\dots \left(i\right)4+0-4g+O+c=0\dots \left(ii\right)$
Also the centre of the circle lies on
$2x-y=3-2g+f=3\dots \left(iii\right)$
Solving equations (i), (ii) and (iii), we get
$g=\frac{3}{2},f=6$ and $c=2$
Therefore the equation of the circle is
$x^2+y^2+3x+12y+2=0$