Question

Compute the orthogonal trajectories of the family F of the curves given by $y^2=c{x}^3$, where c is a arbitrary constant.

• A. $y^2+\frac{x^2}{3}=c$
• B. $y^2+\frac{x^2}{4}=c$
• C. $y^2+\frac{x^3}{3}=c$
• D. None of these

Answer 1

The correct option is D None of these

Given equations of the family of curves is $y^2=c{x}^3$

$c$ is an arbitrary constant.

Now, differentiating the equation with respect to $x$ gives,

$2y\frac{dy}{dx}=3c{x}^2$

$\therefore c=\frac{2y}{3x^2}\left(\frac{dy}{dx}\right)$

Substituting the value of $c$ in the equation of family of curves gives.,

$\frac{3y}{2x}=\frac{dy}{dx}$

Now, to find the orthogonal trajectories we need to replace $\frac{dy}{dx}$ by $-\frac{dx}{dy}$ and integrating gives.,

$\int 3ydy=\int (-2x)dx$

$\frac{3y^2}{2}=-x^2+c$

$2x^2+3y^2=c$

which is equation of the orthogonal trajectories of the given family of curves.