If the chords of contact of tangents from two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angles such that $\frac{x_1{x}_2}{y_1{y}_2}=-\frac{a^m}{b^n}$, where $m$, $n$ are positive integers, find the value of  ${\left(\frac{m+n}{4}\right)}^{10}$.

The equations of the chord of contact of the tangents to the given hyperbola at $(x_1, y_1)$ and $(x_2, y_2)$ are 

   $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1$     …(i)

and $\frac{x{x}_2}{a^2}-\frac{y{y}_2}{b^2}=1$     …(ii)

The slopes of the lines (i) and (ii) are 

   $m_1=\left(\frac{b^2{x}_1}{a^2{y}_1}\right)$ and $m_2=\left(\frac{b^2{x}_2}{a^2{y}_2}\right)$

Since, (i) and (ii) meet at right angles, so

   $\begin{array}{l}{m}_1{m}_2=-1\\ \Rightarrow \left(\frac{b^2{x}_1}{a^2{y}_1}\right)\times \left(\frac{b^2{x}_2}{a^2{y}_2}\right)=-1\end{array}$

$\Rightarrow \frac{x_1{x}_2}{y_1{y}_2}=-\frac{a^4}{b^4}$

Thus, $m=4$ and $n=4$

Hence, the value of ${\left(\frac{m+n}{4}\right)}^{10}={\left(\frac{4+4}{4}\right)}^{10}=2^{10}=1024$.