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Let P be a point on the parabola x 2 = 4 y . If the distance of P from the center of the circle x 2 + y 2 + 6 x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P is

Options

  1. Ax + y + 1 = 0
  2. Bx + 4 y - 2 = 0
  3. Cx + 2 y = 0
  4. Dx - y + 3 = 0

Correct answer

A. x + y + 1 = 0

Step-by-step solution

Let P 2 t , t 2 . Equation normal at P to x 2 = 4 y be y - t 2 = - 1 t x - 2 t . It passes a normal through - 3 , 0 . ⇒ 0 - t 2 = - 1 t - 3 - 2 t ⇒ t 3 + 2 t + 3 = 0 ⇒ t + 1 t 2 - t + 3 = 0 ⇒ t = - 1 So, point P is - 2 , 1 . Equation of a tangent to x 2 = 4 y   at - 2 , 1 . ⇒ x - 2 = 2 y + 1 ∴ x + y + 1 = 0 .

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