Find the circle which passes through $\left(h, k\right)$ and whose center is $\left(a, b\right)$.
Center of the required circle $= \left(a, b\right)$.
The required circle passes through the point $\left(h, k\right)$.
$\therefore \;$ Radius of the required circle $= \sqrt{\left(a - h\right)^2 + \left(b - k\right)^2}$
The equation of the required circle is
$\left(x - a\right)^2 + \left(y - b\right)^2 = \left(\sqrt{\left(a - h\right)^2 + \left(b - k\right)^2}\right)^2$
i.e. $\;$ $\left(x - a\right)^2 + \left(y - b\right)^2 = \left(a - h\right)^2 + \left(b - k\right)^2$
i.e. $\;$ $x^2 + a^2 -2ax + y^2 + b^2 - 2by = a^2 + h^2 - 2ah + b^2 + k^2 - 2bk$
i.e. $\;$ $x^2 + y^2 - 2ax - 2by - h^2 - k^2 + 2ah + 2bk = 0$