Question

Find the equation of the circle whose center lies on the $X$-axis and passing through the points $(-2,3)$ and $(4,5)$.

Answer 1

Given that the center lies on $X$ axis. Let the coordinates of center of circle be $(h,0)$

The distance from center to points $(-2,3)$ and $(4,5)$ is equal

So we get ${(h+2)}^2+{(0-3)}^2={(h-4)}^2+{(0-5)}^2$

$\Rightarrow 12h=28$

$\Rightarrow h=\frac{7}{3}$

$r^2=(x-h{)}^2+(y-k{)}^2$

$r^2=(4-\frac{7}{3}{)}^2+(5{)}^2$

$\therefore$ The radius of circle is $r=\sqrt{25+\frac{25}{9}}=\frac{5\sqrt{10}}{3}$

The equation of circle is ${(x-\frac{7}{3})}^2+y^2=\frac{250}{9}$