Question

The differential equation of all parabolas having their axis of symmetry with the axis of x is:

• A. $y\frac{d^{2}y}{d{x}^2}+(\frac{dy}{dx}{)}^2=0$
• B. $y\frac{d^{2}x}{d{x}^2}+(\frac{dy}{dx}{)}^2=0$
• C. $y\frac{d^{2}y}{d{x}^2}+\left(\frac{dy}{dx}\right)=0$
• D. $y\frac{d^{2}y}{d{x}^2}-\left(\frac{dy}{dx}\right)=0$

Answer 1

Answer: (A)

$y\frac{d^{2}y}{d{x}^2}+(\frac{dy}{dx}{)}^2=0$

$\because$ the symmetry is with x axis
$y^2=4a(x-y)$
$\Rightarrow 2y{y}_1=4a$
$\Rightarrow y{y}_1=2a$
$\Rightarrow y^2=2y{y}_1(x-4)$
$\Rightarrow \frac{y}{2y_1}=x-y$
$\Rightarrow \frac{1}{2}\left[\frac{y_1^2-y_{2}y}{y_1^2}\right]=1$
$\Rightarrow 2y_1^2=y_1^2-y_{2}y$
$\Rightarrow y_1^2+y_{2}y=0$