Hyperbola
Hyperbola
nta_pyq_2025_jan
Grade 11
Question:
Let E : 2 y 2 y and H : . Let the distance between the foci of E and the foci of H be x x + = 1, a > b - = 1 2 2 2 2 a b A B 2\sqrt 3 . If a - A = 2, and the ratio of the eccentricities of E and H is 1 3 , then the sum of the lengths of their latus rectums is equal to:
Step-by-Step Solution
Key Concept: Apply the core result for hyperbola parameters and tangents and simplify using the given constraints.
2 2 x y + = 1 foci are (ae, 0) and (-ae, 0) (3) 2 2 a b 2 2 x y ′ ′ + = 1 foci are (Ae , 0) and (-Ae , 0) 2 2 A B \Rightarrow 2ae = 2\sqrt3 \Rightarrow ae = \sqrt3 ′ ′ and 2Ae = 2\sqrt3 \Rightarrow Ae = \sqrt3 e A ′ \Rightarrow ae = Ae \Rightarrow = ′ e a 1 A \Rightarrow = \Rightarrow a = 3 A 3 a Now a - A = 2 \Rightarrow a - and A = 1 a - 2 \Rightarrow a = 3 3 1 ′ Ae = \sqrt3 \Rightarrow e = and e = \sqrt3 \sqrt3 2 2 2 b = a (1 - e ) 2 b = 6 and B = A ((e ) - 1) = (2) \Rightarrow B = 2 2 2 ′ 2 2 2 2 2 b 2 B sum of LR = + = 8 a A
Correct Answer: 3