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JEE Main20248 Apr 2024Morning ShiftMathematicsHyperbolaActual

Let H: -x^2 a^2 + y^2 b^2 =1 be the hyperbola, whose eccentricity is 3 and the length of the latus rectum is 4 3 . Suppose the point ( , 6), >0 lies on H . If is the product of the focal distances of the point ( , 6) , then ^2+ is equal to

Options

  1. A172
  2. B171
  3. C169
  4. D170

Correct answer

B. 171

Step-by-step solution

aligned & H : y ^2 ~b ^2 - x ^2 a ^2 =1, e = 3 & e = 1+ a ^2 ~b ^2 = 3 a ^2 ~b ^2 =2 & a ^2=2 ~b ^2 & length of L.R. = 2 a ^2 ~b =4 3 & a= 6 aligned P( , 6) lie on y^2 3 - x^2 6 =1 aligned & 12- ^2 6 =1 ^2=66 & Foci =(0, be )=(0,3) &(0,-3) aligned Let d₁ & d₂ be focal distances of P ( , 6) aligned & d ₁= ^2+(6+b e)^2 , ~d ₂= ^2+(6-b e)^2 & ~d ₁= 66+81 , ~d ₂= 66+9 & = d ₁ ~d ₂= 147 75 =105 & ^2+ =66+105=171 aligned SECTION - B

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