Question

The orthogonal trajectories for $y=C{x}^2$ where C is arbitrary constant represent

• A. a family of circles
• B. a family of straight lines
• C. a family of ellipses
• D. a family of hyperbolas

Answer 1

The correct option is C a family of ellipses

$y=C{x}^2$
$\Rightarrow \frac{y}{x^2}=C$,
Differentiating $\frac{x^2{y}^{\prime }-2xy}{x^4}=0$
$\Rightarrow y^{\prime }=\frac{2y}{x}$
Replacing $\frac{dy}{dx}$ by $-\frac{dx}{dy}$, we have $y^{\prime }=\frac{-x}{2y}$
$\Rightarrow 2ydy+xdx=0$
$\Rightarrow y^2+\frac{1}{2}{x}^2=constant$
which represent a family of ellipses.