Question

Assertion :Perpendicular tangents can be drawn to hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ only if $e>\sqrt{2}$. Reason: Locus of point of intersection of perpendicular tangents to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $x^2+y^2=a^2-b^2$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

We know that
"The locus of the point of intersection of two perpendicular tangents to a hyperbola is a circle known as the director circle."
Equation of director circle is
$x^2+y^2=a^2-b^2$
Hence, reason is correct.
Assertion:
For real director circle,
$a^2\ge b^2$
$\Rightarrow \frac{b^2}{a^2}\le 1$
$\Rightarrow e^2-1\le 1$
$\Rightarrow e^2\le 2$
$\Rightarrow e<\sqrt{2}$
Hence, assertion is incorrect.