Find the equation of the circle which passes through the points $\left(h, k\right)$, $\left(k, k\right)$ and $\left(k, h\right)$.
Let the equation of the required circle be $\;$ $x^2 + y^2 + 2gx + 2fy + c = 0$
Since the required circle passes through the points $\left(h, k\right)$, $\left(k, k\right)$ and $\left(k, h\right)$, we have
$h^2 + k^2 + 2hg + 2kf + c = 0$ $\;\;\; \cdots \; (1)$
$k^2 + k^2 + 2kg + 2kf + c = 0$ $\;\;$ i.e. $\;\;$ $2k^2 + 2kg + 2kf + c = 0$ $\;\;\; \cdots \; (2)$
$k^2 + h^2 + 2kg + 2hf + c = 0$ $\;\;\; \cdots \; (3)$
Subtracting equation $(1)$ from equation $(2)$ we get,
$-h^2 + k^2 + 2g \left(k - h\right) = 0$ $\;\;\; \cdots \; (4)$
Subtracting equation $(3)$ from equation $(2)$ we get,
$-h^2 + k^2 + 2f \left(k - h\right) = 0$ $\;\;\; \cdots \; (5)$
We have from equations $(4)$ and $(5)$,
$2 \left(k - h\right) \left(g - f\right) = 0$ $\;\;\; \cdots \; (6)$
If $\;$ $k - h = 0$ $\;$ in equation $(6)$, then the coordinates become trivial $\implies$ $k - h \neq 0$
$\therefore \;$ In equation $(6)$, $\;\;\;$ $g - f = 0$ $\implies$ $g = f$
Substituting $g = f$ in equation $(1)$ gives,
$h^2 + k^2 + 2hf + 2kf + c = 0$ $\;\;\; \cdots \; (7)$
Substituting $g = f$ in equation $(2)$ gives,
$2k^2 + 2kf + 2kf +c = 0$
i.e. $\;$ $2k^2 + 4kf + c = 0$
$\implies$ $c = - 2k^2 - 4kf$ $\;\;\; \cdots \; (8)$
Substituting the value of $c$ from equation $(8)$ in equation $(7)$ gives
$h^2 + k^2 + 2hf + 2kf - 2k^2 -4kf = 0$
i.e. $\;$ $h^2 - k^2 + 2hf - 2 kf = 0$
i.e. $\;$ $h^2 - k^2 + 2f \left(h - k\right) = 0$
i.e. $\;$ $\left(h + k\right) \left(h - k\right) + 2 f \left(h - k\right) = 0$
i.e. $\;$ $h + k + 2f = 0$ $\;\;\;$ $\left[\because \; k - h \neq 0\right]$
$\implies$ $f = \dfrac{- \left(h + k\right)}{2}$
$\therefore \;$ $g = f = \dfrac{- \left(h + k\right)}{2}$
Substituting the value of $f$ in equation $(8)$ gives
$c = - 2k^2 - 4 \times \left[\dfrac{- \left(h + k\right)}{2}\right]k$
i.e. $\;$ $c = - 2k^2 + 2hk + 2k^2$ $\implies$ $c = 2hk$
$\therefore \;$ The equation of the required circle is
$x^2 + y^2 + 2 \times \left[\dfrac{- \left(h + k\right)}{2}\right] x + 2 \times \left[\dfrac{- \left(h + k\right)}{2}\right]y + 2hk = 0$
i.e. $\;$ $x^2 + - \left(h + k\right)x - \left(h + k\right)y + 2hk = 0$