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)

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

Answer 2

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
$y^{2} =4. \frac{y}{2} \frac{dy}{dx}x$
$\Rightarrow y^{2} =2xy \frac{dy}{dx} \Rightarrow \frac{dy}{dx} =\frac{y}{2x}$