Let the coordinate at point of intersection of  normal at $P$ and $Q$ be $(h,k)$ 

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 $\frac{{ – 13}}{2}$