Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade 11
Question:
A normal to the hyperbola $\frac{x^2}{4} - \frac{y^2}{1} = 1$, has equal intercepts on positive $x$ and positive $y$ axis. If the normal touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then the value of $\frac{9}{25}(a^2 + b^2)$ is____.
Step-by-Step Solution
Key Concept: A normal to one conic is tangent to another when it satisfies both the normal condition on the first curve and the tangency condition on the second.
The normal at point $(2\sec\theta, \tan\theta)$ on the curve $\frac{2x}{\sec\theta} + \frac{y}{\tan\theta} = 5$ has slope $-1 - \frac{2\tan\theta}{\sec\theta} = -1$ when $\theta = \frac{\pi}{6}$. The normal equation becomes $y = -x + \frac{5}{\sqrt{3}}$. For this normal to be tangent to an ellipse $a^2 + b^2 = \frac{3}{25}(a^2 + b^2) = 3$, we use the tangency condition that the perpendicular distance from the center equals the semi-major axis, yielding $a^2 + b^2 = 3$.
Correct Answer: 3