Ellipse
Confocal Conics and Latus Rectum
nta_pyq_2023_apr
Grade 11
Question:
Let the eccentricity of an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ be reciprocal to that of the hyperbola $2x^2-2y^2=1$. If the ellipse intersects the hyperbola at right angles, then the square of the length of the latus-rectum of the ellipse is _____.
Step-by-Step Solution
Key Concept: Hyperbola $x^2-y^2=\frac{1}{2}$ is rectangular with $e_H=\sqrt{2}$. So $e_E=\frac{1}{\sqrt{2}}$. Perpendicular intersection implies confocal conics (same foci).
$a=\sqrt{2},\ b^2=1$. LR $=\sqrt{2}$. $(\text{LR})^2=2$.
Correct Answer: 2