Question

The orthogonal trajectory of $y^2=4ax,(a$ being the parameter) is:
(where $c$ is an arbitrary constant)

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

Answer 1

The correct option is D $2x^2+y^2=c$

$y^2=4ax\cdots \left(i\right)\Rightarrow 2y\frac{dy}{dx}=4a\cdots \left(ii\right)$
Eliminating $a$ from equation $\left(i\right)$ and $\left(ii\right),$ we have
$y^2=2y\frac{dy}{dx}x\Rightarrow y=2x\frac{dy}{dx}$
To find the orthogonal trajectory, replacing $\frac{dy}{dx}$ by $-\frac{dx}{dy}$, we get
$\Rightarrow y=2(-\frac{dx}{dy})x$
$\Rightarrow 2xdx+ydy=0$
Integrating both sides, we get
$\Rightarrow x^2+\frac{y^2}{2}=C\Rightarrow 2x^2+y^2=c,(\text{let}c=2C)$