Question

Orthogonal trajectories of family of the curve $x^{2/3}+y^{2/3}=a^{2/3}$, where a is any arbitrary constant, is

• A. $x^{2/3}-y^{2/3}=c$
• B. $x^{4/3}-y^{4/3}=c$
• C. $x^{4/3}+y^{4/3}=c$
• D. $x^{1/3}-y^{1/3}=c$

Answer 1

Answer: (B)

$x^{2/3}+y^{2/3}=a^{2/3}$
or $\frac{2}{3}{x}^{-1/3}+\frac{2}{3}{y}^{-1/3}\frac{dy}{dx}=0$
or $\frac{dy}{dx}=-\frac{x^{-1/3}}{y^{-1/3}}(1)$
Replacing $\frac{dy}{dx}\left(\frac{\pi }{2}-\theta \right)$ by $-\frac{dx}{dy}$, we get
$\frac{dx}{dy}=\frac{x^{-1/3}}{y^{-1/3}}$
or $\int x^{1/3}dx=\int y^{1/3}dy$
or $x^{4/3}-y^{4/3}=c$