Find the equation of the standard rectangular hyperbola whose center is $\left(\dfrac{-1}{2}, \dfrac{-1}{2}\right)$ and which passes through the point $\left(1, \dfrac{1}{4}\right)$.
The equation of standard rectangular hyperbola with center at $\left(h, k\right)$ is:
$\left(x - h\right) \left(y - k\right) = c^2$ $\;\;\; \cdots \; (1)$
Given: Center $= \left(h, k\right) = \left(\dfrac{-1}{2}, \dfrac{-1}{2}\right)$
Putting the values of $h$ and $k$ in equation $(1)$, the required equation of rectangular hyperbola becomes
$\left(x + \dfrac{1}{2}\right) \left(y + \dfrac{1}{2}\right) = c^2$ $\;\;\; \cdots \; (2)$
Equation $(2)$ passes through the point $\left(1, \dfrac{1}{4}\right)$.
$\therefore$ $\;$ We have
$\left(1 + \dfrac{1}{2}\right) \left(\dfrac{1}{4} + \dfrac{1}{2}\right) = c^2$ $\implies$ $c^2 = \dfrac{9}{8}$
$\therefore$ $\;$ The required equation of the rectangular hyperbola [equation $(2)$] is
$\left(x + \dfrac{1}{2}\right) \left(y + \dfrac{1}{2}\right) = \dfrac{9}{8}$