Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade 11
Question:
For hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, let $n$ be the number of points on the plane through which perpendicular tangents are drawn.
if $n = 1$, then $e = \sqrt{2}$
if $n > 1$, then $1 \sqrt{2}$
if $n > 1$, then $e > \sqrt{2}$
Step-by-Step Solution
Key Concept: The director circle of a hyperbola exists only when $a^2 > b^2$, which corresponds to eccentricity $e < \sqrt{2}$.
The locus of points of intersection of perpendicular tangents to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is the director circle $x^2 + y^2 = a^2 - b^2$. Using eccentricity $e^2 = 1 + \frac{b^2}{a^2}$: if $a^2 > b^2$, infinitely many points exist on the circle with $e \sqrt{2}$; if $a^2 = b^2$, exactly one point (the centre) exists with $e = \sqrt{2}$.
Correct Answer: 1,2,3