Question

Find the equation of the circle which passes through the points $(20,3)$, $(19,8)$ and $(2,-9)$. Find its centre and radius.

• A. $(7,-3),13$
• B. $(3,7),13$
• C. $(7,3),13$
• D. $(-3,-7),13$

Answer 1

Answer: (C)

By substitution of coordinates in the general equation of the circle given by
$x^2+y^2+2gx+2fy+c=0$, we have
$40g+6f+c=-409\dots \left(i\right)$
$38g+16f+c=-425\dots \left(ii\right)$
$4g-18f+c=-85\dots \left(iii\right)$
Solving $\left(i\right)$, $\left(ii\right)$ and $\left(iii\right)$, we get
$g=-7,f=-3$ and $c=-111$
Hence, the equation of the circle is
$x^2+y^2-14x-6y-111=0$
or ${\left(x-7\right)}^2+{\left(y-3\right)}^2=13^2$
Therefore, the centre of the circle is $\left(7,3\right)$ and radius is $13$.