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JEE Main202623 January 2026Morning ShiftMathematicsHyperbolaActual

Let the domain of the function f(x)= ₃ ₅ ₇ (9 x-x²-13 ) be the interval ( m , n ) . Let the hyperbola x² a ² - y² ~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 :

Options

  1. A7
  2. B11
  3. C5
  4. D9

Correct 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 .

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