Hyperbola
Hyperbola
nta_pyq_2025_jan
Grade 11
Question:
Let H : 2 y 2 y and H : - be two hyperbolas having length of latus rectums 15\sqrt2 and x x 1 - = 1 2 + = 1 2 2 2 2 a b A B 12\sqrt5 respectively. Let their ecentricities be e = \sqrt and e respectively. If the product of the lengths of their 1 5 2 2 transverse axes is 100\sqrt10, then 25e is equal to ________. 2 2 2
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 2 2 a b (55) 2b 2 = 15\sqrt2 ...(i) a 2 ....(ii) b 5 \sqrt1 + = \sqrt a 2 2 From (i) and (ii) 2 a = 5\sqrt2 and b = 75 2 2 x y - = -1 2 2 A B 2 2A B = 12\sqrt5 ....(iii) Since, product of transverse axis is = 100\sqrt10 (2A) ⋅ (2B) = 100\sqrt10 From (iii) and (iv) 2 A = 150 and B = 5\sqrt5 2 A 11 e 2 = \sqrt1 + = \sqrt 2 B 5 11 2 \therefore 25e = 25 ( ) = 55 2 5 2
Correct Answer: 55