The shortest distance between the line x - y = 1 and the curve x 2 = 2 y is:

Step-by-step solution

Shortest distance between curves is always along common normal. Let P x 0 ,   y 0 be any point on the parabola x 2 = 2 y . Clearly, slope of line x - y = 1 is 1 . Hence, slope of perpendicular line is - 1 . d y d x P = Slope of tangent at P Therefore, - 1 d y d x P = Slope of normal = - 1 ⇒ d y d x P = 1 ⇒ x 0 = 1   ∴ y 0 = 1 2 ⇒ P 1 , 1 2 ∴ Shortest distance = 1 - 1 2 - 1 1 2 + 1 2 = 1 2 2 option 3