Question

The differential equation of the family of parabolas $y^2=4ax$, where $a$ is parameter, is

• A. $\frac{dy}{dx}=\frac{y}{2x}$
• B. $\frac{dy}{dx}=-\frac{y}{2x}$
• C. $\frac{dy}{dx}=-\frac{2x}{y}$
• D. $\frac{dy}{dx}=\frac{2x}{y}$

Answer 1

Answer: (A)

Given equation of parabola is $y^2=4ax$ ... (i)
Differentiating (i) w.r.t. $x$, we get
$2y\frac{dy}{dx}=4a$
$\Rightarrow \frac{dy}{dx}=\frac{2a}{y}\Rightarrow a=\frac{y}{2}\frac{dy}{dx}$
Substituting the value of a in (i), we get
$y^2=4.\frac{y}{2}\frac{dy}{dx}x$
$\Rightarrow y^2=2xy\frac{dy}{dx}\Rightarrow \frac{dy}{dx}=\frac{y}{2x}$