If the area enclosed between the curves y = k x 2 and x = k y 2 , k > 0 , is 1 s q . u n i t . Then k is

Step-by-step solution

The given curves are, y = k x 2 ,   x = k y 2 To find the point of intersection of the curves, put y from the first curve into second, to get x = k k 2 x 4 ⇒ x = 0 or x 3 = 1 k 3 ⇒ x = 1 k Hence, y = 0 or y = k 1 k 2 = 1 k Therefore, point of intersection is 1 k ,   1 k . Hence, the required area is ∫ 0 1 k y 2 - y 1 d x = 1 ⇒ ∫ 0 1 k x k - k x 2 d x = 1 ⇒ 1 k x 3 / 2 3 / 2 - k x 3 3 0 1 k = 1 ⇒ 2 3 k 2 - 1 3 k 2 = 1 ⇒ k 2 = 1 3 ⇒ k = ± 1 3 But, g