Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
Match the following:
(A) If vertices of a rectangle of maximum area inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are extremities of latus rectum. Then eccentricity of ellipse is
(B) If extremities of diameter of the circle $x^2 + y^2 = 16$ are foci of a ellipse, then eccentricity of the ellipse, if its size is just sufficient to contain the circle, is
(C) If normal at point (6, 2) to the ellipse passes through its nearest focus (5, 2), having centre at (4, 2) then its eccentricity is
(D) If extremities of latus rectum of the parabola $y^2 = 24x$ are foci of ellipse and if ellipse passes through the vertex of the parabola, then its eccentricity is
Step-by-Step Solution
Key Concept: Finding eccentricity requires relating geometric constraints (inscribed rectangles, foci positions, latus rectum endpoints) to the fundamental ellipse equation a² = b² + c² and eccentricity e = c/a. Each sub-problem uses specific loci conditions: rectangle vertices at latus rectum extremities, circle containment, normal-focus collinearity, and parabola-ellipse foci correspondence.
For the parabola $y^2 = 24x$, the extremities of the latus rectum are at $(6, ±12)$. For an ellipse with $2be = 24$ and minor axis extremity at $(0,0)$, we have $a = 6$. Using $a^2 = b^2 - b^2e^2$, we get $b^2 = 180$, yielding $e = \sqrt{1 - \frac{36}{180}} = \sqrt{\frac{1}{5}} = \frac{2}{\sqrt{5}}$.
Correct Answer: [A-q] [B-q] [C-s] [D-p]