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JEE Main20246 Apr 2024Evening ShiftMathematicsHyperbolaActual

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and x= 4 13 , respectively. Let the line y- 3 x+ 3 =0 touch this hyperbola at (x₀, y₀ ) . If m is the product of the focal distances of the point (x₀, y₀ ) , then 4 e ^2+ m is equal to ________

Correct answer

0

Step-by-step solution

Given 2 ~b ^2 a =9 and a e = 4 3 equation of tangent y- 3 x+ 3 =0 by equation of tangent Let slope = S = 3 Constant =- 3 By condition of tangency aligned & 6=6 a ^2-9 a & a =2, ~b ^2=9 aligned Equation of Hyperbola is x^2 4 - y^2 9 =1 and for tangent Point of contact is (4,3 3 )= ( x ₀, y ₀ ) Now e = 1+ 9 4 = 13 2 Again product of focal distances aligned & m = ( x ₀ e + a ) ( x ₀ e - a ) & m +4 e ^2=20 e ^2- a ^2 & =20 13 4 -4=61 & aligned (There is a printing mistake in the equation of directrix x= 4 3 . Corrected

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