Question

If the chords of contact of tangents from points $(x_1,y_1)$ and $(x_2,y_2)$ 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, then the value of ${\left(\frac{m+n}{4}\right)}^{10}$ is

Answer 1

The equation of the chord of contact of the tangents to given hyperbola at $(x_1,y_1)$ & $(x_2,y_2)$ are
$\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1\dots \left(1\right)$
and $\frac{x{x}_2}{a^2}-\frac{y{y}_1}{b^2}=1\dots \left(2\right)$

The slopes of $\left(1\right)$ and $\left(2\right)$ are
$m_1=\frac{b^2{x}_1}{a^2{y}_1}$ and $m_2=\frac{b^2{x}_2}{a^2{y}_2}$
Since $\left(1\right)$ and $\left(2\right)$ meet at right angle, so $m_1{m}_2=-1$
$\Rightarrow \left(\frac{b^2{x}_1}{a^2{y}_1}\right)\times \left(\frac{b^2{x}_2}{a^2{y}_2}\right)=-1$
$\Rightarrow \frac{x_1{x}_2}{y_1{y}_2}=\frac{-a^4}{b^4}$
Thus, $m=4$ and $n=4$
Hence ${\left(\frac{m+n}{4}\right)}^{10}=2^{10}=1024$