The shortest distance between the line y = x and the curve y 2 = x – 2 is

Step-by-step solution

y 2 = x – 2 Differentiating w.r.t. x we get, 2 y y ' =   1 ⇒ y ' =     1 2 y For the shortest distance, the tangent at point P   will be parallel to the given line ⇒ y ' | p = 1 2 y 1 = 1 ⇒ y 1 = 1 2 ⇒ x 1 = 2 + 1 2 2 = 9 4 ( ∵ y 1 2 = x 1 - 2 ) The shortest distance between the given curve & the line = The perpendicular distance of point P from the line = 9 4 - 1 2 1 2 + 1 2 = 7 4 2 = 7 4 2