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JEE Main202224 Jun 2022Morning ShiftMathematicsParabolaActual

Let x 2 + y 2 + A x + B y + C = 0 be a circle passing through 0 , 6 and touching the parabola y = x 2 at 2 , 4 . Then A + C is equal to _____

Options

  1. A16
  2. B88 5
  3. C72
  4. D- 8

Correct answer

A. 16

Step-by-step solution

Equation of tangent at 2 ,   4 to the parabola y = x 2 is 1 2 y + 4 = 2 x ⇒ 4 x - y - 4 = 0 Then family of circle touching this tangent at 2 ,   4 is x - 2 2 + y - 4 2 + λ 4 x - y - 4 = 0       . . .   i Since it passes through 0 ,   6 , we get, 4 + 4 + λ - 10 = 0 ⇒ λ = 4 5 So, equation i becomes x 2 + y 2 - 4 5 x - 44 5 y + 84 5 = 0 Hence, A + C = 80 5 = 16

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