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The equation of the common tangent to the parabolas y 2 = 2 x and x 2 = 16 y is

Options

  1. A2 x + y - 2 = 0
  2. Bx - 2 y - 2 = 0
  3. Cx - 2 y + 2 = 0
  4. Dx + 2 y + 2 = 0

Correct answer

D. x + 2 y + 2 = 0

Step-by-step solution

Any tangent to y 2 = 2 x in slope form is y = m x + 1 2 m which is also tangent to x 2 = 16 y ⇒ x 2 = 16 m x + 1 2 m will have equal roots. ⇒ Discriminant of x 2 - 16 m x - 8 m = 0 is zero ⇒ 16 m 2 - 4 × 1 - 8 m = 0 ⇒ m 3 = - 1 8 ⇒ m = - 1 2 ⇒ Equation of the common tangent is y = - x 2 - 1 ⇒ x + 2 y + 2 = 0

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