Question
Let the domain of the function f(x)= _ 3 _ 5 _ 7 (9 x-x^ 2 -13 ) be the interval ( m , n ). Let the hyperbola x^ 2 a ^ 2 - y^ 2 ~b ^ 2 =1 have eccentricity n 3 and the length of the latus rectum 8 ~m 3 . Then b ^ 2 - a ^ 2 is equal to :
JEE Main
Mathematics
Hyperbola
2026
JEE Main 2026 (23 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 7
- B. 11
- C. 5
- D. 9
Answer
A. 7
Step-by-step solution
For f(x) = _3 _5 _7(9x-x^2-13): _7( ) > 0 9x-x^2-13 > 1 (x-2)(x-7) _5( ) > 0 _7(9x-x^2-13) > 1 9x-x^2-13 > 7 (x-4)(x-5) Domain = (4, 5), so m = 4, n = 5. Eccentricity e = n/3 = 5/3, latus rectum = 8m/3 = 32/3. e^2 = 1 + b^2/a^2 b^2/a^2 = 16/9. 2b^2/a = 32/3 32a/9 = 32/3 a = 3, so a^2 = 9, b^2 = 16. b^2 - a^2 = 16 - 9 = 7.
Related: Mathematics — Hyperbola · All PYQ Banks