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

P and Q is ___

(-6.5)
Let coordinate at point of intersection of normals at P and Q be (h,k)
Eq. of normal to the hyperbola  $\frac{x^2}{3^2}-\frac{y^2}{2^2}=1$  at point   $P(3\sec \theta ,2\tan \theta )$  is   $3x\cos \theta +2y\cot \theta =3^2+2^2\mathrm{.........}\left(1\right)$
Similarly, equation of normal to the hyperbola  $\frac{x^2}{3^2}-\frac{y^2}{2^2}$  at point  $Q(3\sec \varphi ,2\tan \varphi )$  is  $3x\cos \varphi +2y\cot \varphi =3^2+2^2\mathrm{..........}\left(2\right)$
 
Given  $\theta +\varphi =\frac{\pi }{2}\Rightarrow \varphi =\frac{\pi }{2}-\theta$  and these passes through  $(h,k)$
 $\Rightarrow 3h\sin \theta +2k\tan \theta =3^2+2^2\mathrm{...............}\left(3\right)$
And  $3h\cos \theta +2k\cot \theta =3^2+2^2\mathrm{.............}\left(4\right)$
Solving (3) and (4), we get
$K=\frac{-39(\cos \theta -\sin \theta )}{6(\cos \theta -\sin \theta )}$
Hence, ordinate of point intersection of normals at P and Q is  $\frac{-13}{2}$