If the differential equation representing the family of all circles touching x -axis at the origin is x 2 - y 2 d y d x = g x y , then g x equals

Step-by-step solution

Equation of circle touching x -axis in origin is x - 0 2 + y - k 2 = k 2 ⇒ x 2 + y 2 - 2 k y = 0       . . . 1 , where k is the radius of the circle. Differentiating the above equation w.r.t. x , we get, 2 x + 2 y d y d x - 2 k d y d x = 0 ⇒ x + y - k d y d x = 0 From equation 1 , ⇒ x + y - x 2 + y 2 2 y d y d x = 0 ⇒ x 2 - y 2 d y d x = 2 x y = g x y ⇒ g x = 2 x