Question

The orthogonal trajectories of the family of curve y = cx^k are given by

• A. x² + cy² = constant
• B. x² + ky² = constant
• C. kx² + y² = constant
• D. x²− ky² = constant

Answer 1

Answer: (B)

x² + ky² = constant

Answer 2

Differentiating the given relation we have,

$\frac{dy}{dx}$ = ckx^k−1 ⇒ c = $\frac{1}{k}$ x^1−k $\frac{dy}{dx}$.

Putting this value in the given equation we have

y = $\frac{1}{k}$x^1−k $\frac{dy}{dx}$x^k = $\frac{1}{k}$x $\frac{dy}{dx}$.

Replacing $\frac{dy}{dx}$ by – $\frac{dx}{dy}$, we get

y = − $\frac{1}{k}$x $\frac{dx}{dy}$ ⇒ ky dy + x dx = 0 ⇒ ky² + x² = constant