Find the equation of a circle whose radius is 5 units and passes through two points on the x axis which are…

As, given the circle is passing through points $\left(4, 0\right)$ and $\left(-4, 0\right)$,

So, the centre must lie on $y-$axis.

Let, the coordinate of the centre is $\left(0, c\right)$,

Hence, the radius, $\sqrt{{\left(0-4\right)}^2+{\left(c-0\right)}^2}=5 \Rightarrow c=\pm 3$,

$\Rightarrow$the co ordinate of the centre is $\left(0, 3\right) \mathrm{or} \left(0,-3\right)$.

So, the equation of such circles are${\left(x-0\right)}^2+{\left(y-3\right)}^2=5^2 \Rightarrow x^2+y^2-6y-16=0$

or

${\left(x-0\right)}^2+{\left(y+3\right)}^2=5^2 \Rightarrow x^2+y^2+6y-16=0$