JEE Main 2024 Mathematics question
Consider a hyperbola H having centre at the origin and foci on the x-axis. Let C be the circle touching the hyperbola H and having the centre at the origin. Let C be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq. units) of C and C are 36π and 4π, respectively, then the length (in units) of latus rectum of H is
Answer: (A)
Solution
C₁ touches at the vertices, so a = 6; C₂ has radius ae − a = 2 ⇒ ae = 8. Then b² = 64 − 36 = 28 and the latus rectum is 2b²/a = 56/6 = 28/3.