A standard rectangular hyperbola has its vertices at $\left(5, 7\right)$ and $\left(-3, -1\right)$. Find its equation and asymptotes.
Let the required equation of the general rectangular hyperbola be
$\left(x - h\right) \left(y - k\right) = c^2$ $\;\;\; \cdots \; (1)$
The vertices of the rectangular hyperbola are $\left(5, 7\right)$ and $\left(-3, -1\right)$.
Let the center of the rectangular hyperbola be $\left(h, k\right)$.
The midpoint of the vertices of a rectangular hyperbola is its center.
$\therefore$ $\;$ We have, $\;$ $h = \dfrac{5 - 3}{2} = 1$ $\;$ and $\;$ $k = \dfrac{7 - 1}{2} = 3$
$\therefore$ $\;$ The center of the rectangular hyperbola is $\left(h, k\right) = \left(1, 3\right)$
Substituting the values of $h$ and $k$ in equation $(1)$ we have,
$\left(x -1\right) \left(y - 3\right) = c^2$ $\;\;\; \cdots \; (2)$
The vertices $\left(5, 7\right)$ and $\left(-3, -1\right)$ lie on the hyperbola.
$\therefore$ $\;$ We have from equation $(2)$,
$\left(5 - 1\right) \left(7 - 3\right) = c^2$ $\implies$ $c^2 = 16$
Substituting the value of $c^2$ in equation $(2)$, the required equation of rectangular hyperbola is
$\left(x -1\right) \left(y - 3\right) = 16$
The equation of rectangular hyperbola differs from the combined equation of asymptotes by a constant.
$\therefore$ $\;$ The combined equation of the asymptotes is $\;$ $\left(x -1\right) \left(y - 3\right) = 0$
$\therefore$ $\;$ The equations of the asymptotes are
$x -1 = 0$ $\;$ and $\;$ $y - 3 = 0$