Question

The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, $x^{2} \sec ^{2} \alpha-$ $y^{2} \operatorname{cosec}^{2} \alpha=1, \alpha \in(0, \pi / 4)$, is

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (D)

$\frac{x^{2}}{\cos ^{2} \alpha}-\frac{y^{2}}{\sin ^{2} \alpha}=1$
Locus of perpendicular tangents is director circle.
$x^{2}+y^{2} =a^{2}-b^{2} $
or $ x^{2}+y^{2} =\cos ^{2} \alpha-\sin ^{2} \alpha=\cos 2 \alpha$
But $0 < \alpha < \frac{\pi}{4}$
$0 < 2 \alpha < \frac{\pi}{2}$
So there are infinite points.