Number of straight lines which are tangent to  $y^2=8\sqrt{5}x$  and normal to  $x^2-9y^2=9$ is equal to

Let the normal to the hyperbola be  $3x\cos \theta +y\cot \theta =10$…………(i)
and tangent to the parabola be  $y=mx+\frac{2\sqrt{5}}{m}$ ……………….(ii)
Comparing (i) & (ii)
$\frac{-m.\sec \theta }{3}=\tan \theta =\frac{1}{\sqrt{5}m}\Rightarrow \sec \theta =-\frac{3}{\sqrt{5}{m}^2}$  and  $\tan \theta =\frac{1}{\sqrt{5}m}$
Now,  ${\sec }^2\theta =1+{\tan }^2\theta$
 $\Rightarrow \frac{9}{5m^4}=1+\frac{1}{5m^2}$$\Rightarrow 5m^4+m^2-9=0$
$m^2=\frac{-\pm \sqrt{181}}{10}$ the two real values