Question

The orthogonal trajectories for $y=C{x}^{3/2}$ where C is arbitrary constant is given by

• A. $x^2=2y=k$
• B. $2x^2+3y^2=k$
• C. $y^2=2x+k$
• D. $3x^2+2y^2=k$

Answer 1

The correct option is B $2x^2+3y^2=k$

$y{x}^{-3/2}=C$
Differentiating, $y^{\prime }{x}^{-3/2}-\frac{3}{2}{x}^{-5/2}y=0$.
Replace $\frac{dy}{dx}by-\frac{dx}{dy}$,
we have, $1+\frac{3}{2}{x}^{-1}y\frac{dy}{dx}=0$
$\Rightarrow ydy=-\frac{2}{3}xdx$
$\Rightarrow \frac{y^2}{2}=-\frac{1}{3}{x}^2+counst$
$\Rightarrow 2x^2+3y^2=const$