Question
Let $$P\left( {a\,\sec \,\theta ,\,b\,\tan \,\theta } \right)$$ and $$Q\left( {a\,\sec \,\phi ,\,b\,\tan \,\phi } \right),$$ where $$\theta + \phi = \frac{\pi }{2},$$ be two points on the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1.$$ If $$\left( {h,\,k} \right)$$ is the point of intersection of the normal at $$P$$ and $$Q,$$ then $$k$$ is equal to :
A.
$$\frac{{{a^2} + {b^2}}}{a}$$
B.
$$ - \left( {\frac{{{a^2} + {b^2}}}{a}} \right)$$
C.
$$\frac{{{a^2} + {b^2}}}{b}$$
D.
$$ - \left( {\frac{{{a^2} + {b^2}}}{b}} \right)$$
Answer :
$$ - \left( {\frac{{{a^2} + {b^2}}}{b}} \right)$$
Solution :
KEY CONCEPT :
Equation of the normal to the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$$ at the point $$\left( {a\,\sec \,\alpha ,\,b\,\tan \,\alpha } \right)$$ is given by $$ax\,\cos \,\alpha + by\,\cot \,\alpha = {a^2} + {b^2}$$
Normal at $$\theta ,\,\phi $$ are \[\left\{ \begin{array}{l}
ax\,\cos \,\theta + by\,\cot \,\theta = {a^2} + {b^2}\\
ax\,\cos \,\theta + by\,\cot \,\phi = {a^2} + {b^2}
\end{array} \right.\]
where $$\phi = \frac{\pi }{2} - \theta $$ and these pass through $$\left( {h,\,k} \right)$$
$$\eqalign{
& \therefore ah\,\cos \,\theta + bk\,\cot \,\theta = {a^2} + {b^2} \cr
& \,\,\,\,ah\,\sin \,\theta + bk\,\tan \,\theta = {a^2} + {b^2} \cr} $$
Eliminating $$h,\,bk\left( {\cot \,\theta \,\sin \,\theta - \tan \,\theta \cos \,\theta } \right)$$
$$ = \left( {{a^2} + {b^2}} \right)\left( {\sin \,\theta - \cos \,\theta } \right)\,\,\,\,{\text{or }}\,k = - \frac{{\left( {{a^2} + {b^2}} \right)}}{b}$$