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The orthogonal trajectory of y 2 = 4 a x (where a being parameter) is

Options

  1. A2 x 2 + y 2 = k
  2. B2 x 2 - y 2 = k
  3. Cx 2 + y 2 = k
  4. Dx 2 - y 2 = k

Correct answer

A. 2 x 2 + y 2 = k

Step-by-step solution

y 2 = 4 a x ⇒     2 y d y d x = 4 a So, the differential equation of y 2 = 4 a x is y 2 = 2 y d y d x x ⇒       y = 2 x d y d x Replace d y d x by - d x d y in above equation, we get, y = - 2 x d x d y ⇒     y d y = - 2 x d x ⇒     ∫ y d y = - 2 ∫ x d x ⇒   y 2 2 = - x 2 + c ⇒     y 2 = - 2 x 2 + 2 c ⇒       2 x 2 + y 2 = k is the required orthogonal trajectory.

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