If the curves x = y 4 and x y = k cut at right angles, then 4 k 6 is equal to ___ .

Step-by-step solution

Given, x = y 4   and x y = k By solving equations, we get y 5 = k       … 1 Also x = y 4 ⇒ 1 = 4 y 3 d y d x ⇒ d y d x = 1 4 y 3 for x y = k ⇒ x = k y ⇒ 1 = - k y 2 · d y d x ⇒ d y d x = - y 2 k ∵ Curve cut orthogonally ⇒ 1 4 y 3 × - y 2 k = - 1 ⇒ y = 1 4 k ∴ from 1   y 5 = k ⇒ 1 4 k 5 = k ⇒ 4 = 4 k 6