Q. The differential equation of the family of parabolas with focus at the origin and the $x$-axis as axis, is
A) $y\ \left( \ \frac{dy}{dx}\ \right)^{2} + 4x\frac{dy}{dx} = 4y$
B) $- y\ \left( \ \frac{dy}{dx}\ \right)^{2} = 2x\frac{dy}{dx} - y$
C) $y\ \left( \ \frac{dy}{dx}\ \right)^{2} + y = 2xy\frac{dy}{dx}$
D) $y\ \left( \ \frac{dy}{dx}\ \right)^{2} + 2xy\frac{dy}{dx} + y = 0$
Explanation
(b)
Equation of family of parabolas with focus at (0, 0) and $x$-axis as axis is
$y^{2} = 4a(x + a)$ ….(i)
On differentiating Eq. (i), we get
$2yy_{1} = 4a$, putting the value of $a$ in Eq. (i)
$\Rightarrow y^{2} = 2yy_{1}\left( \ x + \frac{yy_{1}}{2}\ \right)$
$\Rightarrow y = 2xy_{1} + yy_{1}^{2}$
$\Rightarrow y\ \left( \ \frac{dy}{dx}\ \right)^{2} + 2x\ \frac{dy}{dx} = y$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.