If $\mathrm{P}(3\mathrm{sec\theta } ,2\mathrm{tan\theta })$ and $\mathrm{Q}(3\mathrm{sec\varphi } ,2\tan \mathrm{\varphi })$ where $\mathrm{\theta }+\mathrm{\varphi }=\frac{\mathrm{\pi }}{2},$ be two distinct points on the hyperbola $\frac{{\mathrm{x}}^2}{9}-\frac{{\mathrm{y}}^2}{4}=1.$ Then the ordinate of the points of intersection of the normals at P and Q is 

Given hyperbola $\frac{{\mathrm{x}}^2}{9}-\frac{{\mathrm{y}}^2}{4}=1$

$\Rightarrow {\mathrm{a}}^2=9, {\mathrm{b}}^2=4$

$\mathrm{Equation} \mathrm{of} \mathrm{normal} \mathrm{at} \mathrm{P}(3\mathrm{sec\theta }, 2\mathrm{tan\theta }) \mathrm{is}$

$$\begin{gathered} \frac{3\mathrm{x}}{\mathrm{sec\theta }}+\frac{2\mathrm{y}}{\mathrm{tan\theta }}=13 \\ \Rightarrow 3\mathrm{x} \mathrm{cos\theta }+2\mathrm{y} \mathrm{cot\theta }=13 \to \left(1\right) \\ \mathrm{Equation} \mathrm{of} \mathrm{normal} \mathrm{at} \cancel{0} (3\mathrm{se}\varnothing , 2\tan \varnothing ) \mathrm{is} \\ 3\mathrm{x} \cos \varnothing +2\mathrm{y} \cot \varnothing =13 \\ 3\mathrm{x} \mathrm{sin\theta }+2\mathrm{y} \mathrm{tan\theta }=13 \to \left(2\right) (\therefore \varnothing =\frac{\mathrm{\pi }}{2}-\mathrm{\theta }) \\ \left(1\right)\times \mathrm{sin\theta }-\left(2\right)\times \mathrm{cos\theta } \\ \Rightarrow 2\mathrm{y}(\mathrm{cos\theta }-\mathrm{sin\theta })=13(\mathrm{sin\theta }-\mathrm{cos\theta }) \\ \Rightarrow 2\mathrm{y}=-13 \\ \Rightarrow \mathrm{y}=\frac{-13}{2}. \end{gathered}$$