Find the equation of the circle which passes through two points on the x-axis which are at distances 4unit from the origin and whose radius is 5.

There are two circles which pass through two points $AandA\text{'}$ at a distance of 4 units from the origin and whose radius is 5. The centre of these circles lie on y-axis
In  $\Delta OAC,A{C}^2=O{A}^2+O{C}^2$
$\Rightarrow 5^2=4^2+O{C}^2\Rightarrow OC=3$
So the co-ordinates of the centres of the required circles are $C(0,3)andC\text{'}(0,-3)$.
Hence the equations of the required circles are  ${(x-0)}^2+{(y\mp 3)}^2=5^2$
$\Rightarrow x^2+y^2\mp 6y-16=0$