Question

If a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose centre is $C$ meets the transverse axis in P and the conjugate axis in Q, then the value of $\frac{a^2}{C{P}^2}-\frac{b^2}{C{Q}^2}$ equals :

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is C $1$

The equation of the tangent at $(asec\theta ,btan\theta )$ to the hyperbola is
$\frac{x}{a}sec\theta -\frac{y}{b}tan\theta =1$
Putting $y=0,$ we get point P $(\frac{a}{sec\theta },0)$
Putting $x=0,$ we get point $Q(0,\frac{b}{tan\theta })$
$\therefore C{P}^2=\frac{a^2}{se{c}^2\theta },C{Q}^2=\frac{b^2}{ta{n}^2\theta }$
Therefore, $\frac{a^2}{C{P}^2}-\frac{b^2}{C{Q}^2}=se{c}^2\theta -ta{n}^2\theta =1$
Hence, option 'C' is correct