Ellipse
Ellipse
nta_pyq_2025_jan
Grade 11
Question:
Let E : 2 2 y be an ellipse. Ellipses E 's are constructed such that their centres and eccentricities are x 1 + = 1 1 9 4 same as that of E , and the length of minor axis of E is the length of major axis of E 1 i i+1 (i \ge 1) . If A is the area i of the ellipse E , then , is equal to 5 \infty i (\sum Ai ) \pi i=1
Step-by-Step Solution
Key Concept: Apply the core result for ellipse parameters and tangents and simplify using the given constraints.
(54) 2 2 x y 4 \sqrt5 E1 = + \Rightarrow e = \sqrt1 - = 9 4 9 3 2 2 x y E2 : + = 1 2 a 4 2 2 \sqrt5 a 5 a e = = \sqrt1 - \Rightarrow = 1 - 3 4 9 4 16 2 a = 9 2 2 x y E2 : + = 1 16 4 9 2 2 x y E3 : + = 1 16 2 b 9 2 \sqrt5 b 64 2 e = = 1 - \Rightarrow b = 3 ⎷ 16 81 9 2 2 x y E3 = + = 1 16 64 9 81 A1 = \pi \times 3 \times 2 \Rightarrow 6\pi 4 8\pi A2 = \pi \times \times 2 = 3 3 4 8 32\pi A3 = \pi \times \times = 3 9 81 \infty 8\pi 32\pi 6\pi 54\pi \sum Ai = 6\pi + + + \ldots.\infty \Rightarrow \Rightarrow 4 3 81 5 i=1 1 - 9 \infty 5 5 54\pi \therefore \sum Ai \Rightarrow \times = 54 \pi \pi 5 i=1
Correct Answer: 54