The chord PQ of the rectangular hyperbola $\mathrm{xy}={\mathrm{a}}^2$ meets the axis of x at A; C is the midpoint of PQ and O is the origin. Then the $∆$ACO is

$Given hyperbola xy=a^2$ and O=(0,0)

Let $\mathrm{C}\left(h,k\right)$ is midpoint of $\overline{\mathrm{PQ}}$

Now Equation of chord PQ is 

 ${\mathrm{S}}_1={\mathrm{S}}_{11}$

$$\begin{gathered} \Rightarrow hy+kx-2a^2=hk-a^2 \\ \Rightarrow hy+kx=hk+a^2 \\ \Rightarrow hy+kx=2hk \\ \Rightarrow \frac{x}{h}+\frac{y}{k}=2 \\ Since chord meets x-axis then y=0 \\ \Rightarrow x=2h \\ and A=(2h,0) \\ Now OC=\sqrt{h^2+k^2} \\ and AC=\sqrt{h^2+k^2} \end{gathered}$$

So, ΔOCA is isosceles with OC=AC.