Since, equation of normal to the hyperbola $\,\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ At point $\left( {{x_1},{y_1}} \right)$ is
$\frac{{{a^2}x}}{{{x_1}}} + \frac{{{b^2}y}}{{{y_1}}} = {a^2} + {b^2}$ therefore equation of normal to the hyperbola $\frac{{{x^2}}}{{{3^2}}} - \frac{{{y^2}}}{{{2^2}}} = 1$
at point $P\left( {3\,\sec \theta ,2\,\tan \theta } \right)$ is
$\frac{{{3^2}x}}{{3\,\sec \theta }} + \frac{{{2^2}y}}{{2\tan \theta }} = {3^2} + {2^2}$
$ \Rightarrow \boxed{3x\,\cos \theta + 2y\cot \theta = {3^2} + {2^2}} ......\left( 1 \right)$
Similarly , equation of normal to the hyperbola $\frac{{{x^2}}}{{{3^2}}} - \frac{{{y^2}}}{{{2^2}}}$ at point
$Q\left( {3\,\sec \phi ,2\,\tan \phi } \right)$ is
$\frac{{{3^2}x}}{{3\,\sec \phi }} + \frac{{{2^2}y}}{{2\tan \phi }} = {3^2} + {2^2}$
$ \Rightarrow \boxed{3x\,\cos \phi + 2y\cot \phi = {3^2} + {2^2}} ......\left( 2 \right)$
Given $\theta + \phi = \frac{\pi }{2} \Rightarrow \phi = \frac{\pi }{2} - \theta $ and these passes through $(h,k)$
$\therefore $ Frem eq. $(2)$
$ \Rightarrow \boxed{3h\,\sin \theta + 2k\tan \theta = {3^2} + {2^2}} ......\left( 3 \right)$
and $\boxed{3h\,\cos \theta + 2k\cot \theta = {3^2} + {2^2}} ......\left( 4 \right)$
Commparing equation $(3) \;and \;(4)$, we get
$3h\,\cos \theta + 2k\cot \theta = 3h\,\sin \theta + 2k\tan \theta $
$3h\,\cos \theta - 3h\sin \theta = 2k\tan \theta - 2k\cot \theta $
$3h\left( {\cos \theta - \sin \theta } \right) = 2k\left( {\tan \theta - \cot \theta } \right)$
$3h\left( {\cos \theta - \sin \theta } \right)$
$ = 2k\frac{{\left( {\sin \theta - \cos \theta } \right)\left( {\sin \theta + \cos \theta } \right)}}{{\sin \theta \cos \theta }}$
or $3h = \frac{{ - 2k\left( {\sin \theta \cos \theta } \right)}}{{\sin \theta \cos \theta }} ......\left( 5 \right)$
Now, putting the value of equation $(5)$ in eq.$(3)$
$\frac{{ - 2k\left( {\sin \theta \cos \theta } \right)\sin \theta }}{{\sin \theta \cos \theta }} + \,2k\tan \theta = {3^2} + {2^2}$
$ \Rightarrow \,2k\tan \theta - 2k + 2k\tan \theta = 13$
$ - 2k = 13 \Rightarrow k = \frac{{ - 13}}{2}$
Hence, ordinate of point of intersection of the at $P$ and $Q$ is $\frac{{ - 13}}{2}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.