Question

Show that the points (5, 5), (6, 4), (−2, 4) and (7, 1) all lie on a circle, and find its equation, centre and radius.

Answer 1

Let the required equation of the circle be $x^2+y^2+2gx+2fy+c=0$. ...(1)

It is given that the circle passes through (5, 5), (6, 4), (−2, 4).
∴ $50+10g+10f+c=0$ ...(2)
$52+12g+8f+c=0$ ...(3)
$20-4g+8f+c=0$ ...(4)

Solving (2), (3) and (4):
$g=-2,f=-1,c=20$

Thus, the equation of the circle is $x^2+y^2-4x-2y-20=0$. ...(5)

We see that the point (7, 1) satisfies equation (5).

Hence, the points (5, 5), (6, 4), (−2, 4) and (7, 1) lie on the circle.

Also, centre of the required circle = $\left(2,1\right)$
Radius of the required circle = $\sqrt{4+1+20}=5$