Question

The orthogonal trajectories of the family of curves $a^{n-1}y=x^n$ are given by

• A. $x^n+n^{2}y=$ constant
• B. $n{y}^2+x^2=$ constant
• C. $n^{2}x+y^n=$ constant
• D. $n^{2}x-y^n=$ constant

Answer 1

The correct option is B $n{y}^2+x^2=$ constant

Differentiating, we have $a^{n-1}\frac{dy}{dx}=n{x}^{n-1}\Rightarrow a^{n-1}=n{x}^{n-1}\frac{dx}{dy}$
Putting this value in the given equation, we have$n{x}^{n-1}\frac{dx}{dy}y=x^n$
Replacing $\frac{dy}{dx}$ by $-\frac{dx}{dy}$ we have $ny=-x\frac{dx}{dy}$
$\Rightarrow nydy+xdx=0\Rightarrow n{y}^2+x^2=$ constant. Which is the required family of orthogonal trajectories.