Tangents of Rectangular hyperbola

Question

P,Q,R are points on a rectangular hyperbola, and PQ perpendicular to PR. Prove that the tangent at P is perpendicular to QR.

Answer 1

Using parametric points, let P$(t_1)$,Q$(t_2)$ and R$(t_3)$ are the required points.

Slope of PQ is $-\frac{1}{t_1t_2}$ and of PR is $-\frac{1}{t_1t_3}$

Given that $-\frac{1}{t_1t_2}$=$t_1t_3$, $\because$ PQ $\perp$ PR

$t_1^2t_3t_2=-1$ $\implies$ $t_1^2=-\frac{1}{t_3t_2}$ $\cdots$(1)

Slope of tangent at P is $-\frac{1}{t_1^2}$ ($m_1$) and slope of QR is $-\frac{1}{t_2t_3}$ ($m_2$)

Now, we have to prove that $m_2$=-$\frac{1}{m_1}$, Tangent at P is $\perp$ to QR.

$t_1^2=-\frac{1}{t_3t_2}$

Which is true!