Find the equation of the circle which passes through two points on the X axis which are at distance 4 from the origin and whose radius is 5.
Let the required circle pass through points $A$ and $B$.
$A$ and $B$ are such that their distance from the origin is $4$ units.
$\therefore$ $\;$ Let $A = \left(-4,0\right)$ and $B = \left(4,0\right)$
From the figure, the center of the circle will lie on the Y axis.
Let the center of the circle be $C = \left(0,b\right)$.
Radius of the circle $= AC = BC = 5$ units (given)
Now, $AC = \sqrt{\left(0 + 4\right)^2 + \left(b - 0\right)^2} = 5$
i.e. $16 + b^2 = 25$ $\implies$ $b^2 = 9$ $\implies$ $b = \pm 3$
$\therefore$ $\;$ The centers of the required circles are $\left(0, \pm 3\right)$
$\therefore$ $\;$ Equations of the required circles are $\hspace{1em}$ $\left(x - 0\right)^2 + \left(y \mp 3\right)^2 = 5^2$
i.e. $x^2 + y^2 \mp 6y + 9 = 25$
i.e. $x^2 + y^2 \mp 6y - 16 = 0$