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JEE Main202227 Jul 2022Morning ShiftMathematicsParabolaActual

Let P a , b be a point on the parabola y 2 = 8 x such that the tangent at P passes through the centre of the circle x 2 + y 2 - 10 x - 14 y + 65 = 0 . Let A be the product of all possible values of a and B be the product of all possible values of b . Then the value of A + B is equal to

Options

  1. A0
  2. B25
  3. C40
  4. D65

Correct answer

D. 65

Step-by-step solution

Given P a , b is point on y 2 = 8 x , such that tangent at P pass through centre of x 2 + y 2 - 10 x - 14 y + 65 = 0 i.e. 5 , 7 Now tangent at P 2 t 2 , 4 t will be t y = x + 2 t 2 and it pass through 5 , 7 So, 7 t = 5 + 2 t 2 ⇒ t = 1 , t = 5 2 So, point P 2 t 2 , 4 t ⇒ 2 , 4 when t = 1 and 25 2 , 10 when t = 5 2 Now product of all possible value of a will be A = 2 × 25 2 = 25 And product of all possible value of b will be B = 4 × 10 = 40 ∴     A + B = 65

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