Question

The orthogonal trajectories of the family of curves a^{n – 1} y = xⁿ are given by –

• A. xⁿ + n²y = const
• B. ny² + x² = const
• C. n²x + yⁿ = const
• D. n²x – yⁿ = const

Answer 1

Answer: (B)

ny² + x² = const

Answer 2

Differentiating, we have (see theory)

a^n–1 $\frac{dy}{dx}$ = nx^{n – 1} Ž a^{n – 1} = n x^{n – 1} $\frac{dx}{dy}$

Putting this value in the given equation, we have

nx^{n – 1} $\frac{dx}{dy}$ y = xⁿ

Replacing $\frac{dy}{dx}$ by – $\frac{dx}{dy}$, we have ny = –x $\frac{dx}{dy}$

Ž ny dy + x dx = 0 Ž ny² + x² = const.

Which is the required family of orthogonal trajectories