Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes a triangle of constant area.
Let the rectangular hyperbola be: $\;$ $xy = c^2$ $\;\;\; \cdots \; (1)$
The asymptotes of the rectangular hyperbola given by equation $(1)$ are the $X$ and $Y$ coordinate axes.
Any point $P \left(t\right)$ on equation $(1)$ is $\;$ $P \left(t\right) = P \left(ct, \dfrac{c}{t}\right)$
Equation of tangent at point $P \left(t\right)$ is
$x + yt^2 = 2ct$ $\;\;\; \cdots \; (2)$
Putting $y = 0$ in equation $(2)$, we get the coordinates of point $A$ as $A \left(2ct, 0\right)$.
Putting $x = 0$ in equation $(2)$, we get the coordinates of point $B$ as $B \left(0, \dfrac{2c}{t}\right)$
$\begin{aligned}
\therefore \; \text{Area of } \triangle AOB & = \dfrac{1}{2} \times OA \times OB \\\\
& = \dfrac{1}{2} \times 2 ct \times \dfrac{2c}{t} \\\\
& = 2c^2 = \text{constant}
\end{aligned}$
$\therefore$ $\;$ The tangent at any point to the rectangular hyperbola forms with the asymptotes a triangle of constant area.
Hence proved.