Ellipse
Ellipse
nta_pyq_2025_jan
Grade 11
Question:
Let the ellipse E : xa 2 + yb 2 = 1, a > b and E 2 : xA 2 + yB 2 = 1, A < B have same eccentricity 1 1 32 . Let the product of their lengths of latus rectums be , and the distance between the foci of E 1 be 4. If 3 3 E 1 and E 2 meet at A, B, C and D , then the area of the quadrilateral ABCD equals : 6
12 5
6 6 6
18 5 6
24 5 2
Step-by-Step Solution
Key Concept: Apply the core result for ellipse parameters and tangents and simplify using the given constraints.
2ae = 4 (4) \Rightarrow a = 2\sqrt3 2 b 1 2 \Rightarrow 1 - = \Rightarrow b = 8 12 3 2 2 2b 2A 32 \times = a B \sqrt3 \Rightarrow 2 2 \times 8 2A 32 \times = 2\sqrt 3 B \sqrt3 \Rightarrow 2 A 2 = 2 \Rightarrow A = 2B B 2 A 1 1 - = B 3 \Rightarrow 2 B = 3 \Rightarrow A = 6 2 2 y E1 : x 12 + 8 = 1 ....(i) 2 2 y ...(ii) x E1 : + = 1 6 9 On solving (i) \& (ii) \sqrt6 6 -\sqrt6 6 \sqrt6 -6 (x, y) = ( , ),( , ),( , ), \sqrt5 \sqrt5 \sqrt5 \sqrt5 \sqrt5 \sqrt5 -\sqrt6 -6 ( , ) \sqrt5 \sqrt5 24\sqrt6 Four points are vertices of rectangle area = 5
Correct Answer: 4