Question

Find the equations to the circles which pass through the point :
(1, 1), (2, -1), and (3, 2).

Answer 1

Let the equation of circle be

$x^2+y^2+2gx+2fy+c=0$

The circle passes through the three points, so they must satisfy the circle's equation

For point 1

$2+2g+2f+c=0$ ....... (1)

For point 2

$5+4g-2f+c=0$ ......... (2)

For point 3

$13+6g+4f+c=0$ ........ (3)

Solving above equations, we get

$g=\frac{-5}{2},f=\frac{-1}{2},c=4$

Then, the equation of circle becomes

$x^2+y^2-5x-y+4=0$