Step 1:

Let the equation of the rectangular hyperbola be $xy=c^2$ and let $P(x_1,y_1)$ be any point on it.

Therefore $x_1y_1=c^2$

The tangent at $P$ is $\large\frac{1}{2}$$(xy_1+yx_1)=c^2$

(i.e) $xy_1+yx_1=2c^2$

Step 3:

It intersects the $x$ and $y$-axes at $A$ and $B$ respectively.

The $x$-coordinates of $A$ and $y$-coordinates of $B$ are obtained by $y=0$ and $x=0$ in the equation of the tangent.$A$ and $B$ are the points $A(\large\frac{2c^2}{y_1},$$0),B(0,\large\frac{2c^2}{x_1})$

Step 4:

Now $OA=\large\frac{2c^2}{y_1},$$OB=\large\frac{2c^2}{x_1}$ and $\Delta AOB$ is rightangled at $O$.

Its area=$\large\frac{1}{2}$$OA\times OB$

$\qquad\;\;=\large\frac{1}{2}$$\times\large\frac{2c^2}{y_1}\times $$\large\frac{2c^2}{x_1}$

$\Rightarrow \large\frac{2c^4}{x_1y_1}$

We know that $x_1y_1=c^2$

$\Rightarrow \large\frac{2c^4}{c^2}$

$\Rightarrow 2c^2$

Therefore the area of the $\Delta$ formed is a constant.