Q. The differential equation of all coaxial parabola $y^{2} = 4a(x - b)$, where $a$ and $b$ are arbitrary constants, is
A) $y\ \frac{d^{2}y}{dx^{2}} + \frac{dy}{dx} = 1$
B) $y\ \frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{2} = 1$
C) $y\ \frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{2} = 0$
D) $y\ \frac{d^{2}y}{dx^{2}} + \frac{dy}{dx} = 0$
Explanation
(c)
Given, $y^{2} = 4a(x - b)$
$\Rightarrow$ $2y\ \frac{dy}{dx} = 4a$
$\Rightarrow$ $2y\frac{d^{2}y}{dx^{2}} + 2\left( \frac{dy}{dx} \right)^{2} = 0$
$\Rightarrow$ $y\frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{2} = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.