Question

$P$ and $Q$ are the variable points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that the slope of the chord is a constant $m$. Prove that the locus of the point of intersection of the tangents at $P$ and $Q$ is $b^{2}x=m{a}^{2}y$

Answer 1

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Let point of intersection be $A(h,k)$

Equation of chord of contact PQ

$\Rightarrow \frac{xh}{a^2}-\frac{yk}{b^2}=1$

Slope of chord m$=\frac{h/a^2}{k/b^2}=\frac{h{b}^2}{k{a}^2}$

$\Rightarrow h{b}^2=mk{a}^2$

$\Rightarrow x{b}^2=\left(m{a}^2\right)y$.