The differential equation corresponding to the family of parabolas $y^2=4 a(x+a)$, where $a$ is the parameter, is

• A

  $y\left(\frac{d y}{d x}\right)^2+2 x \frac{d y}{d x}+y=0$
• B

  $y\left(\frac{d y}{d x}\right)^2-2 x \frac{d y}{d x}-y=0$
• Correct answer
  C

  $y\left(\frac{d y}{d x}\right)^2+2 x\left(\frac{d y}{d x}\right)-y=0$
• D

  $y=2 x \frac{d y}{d x}$

Solution

Given family of parabola is
From Eqs. (i) and (ii), $$ \begin{aligned} & y^2=4 x\left(\frac{1}{2} y \frac{d y}{d x}\right)+4\left(\frac{1}{2} y \frac{d y}{d x}\right)^2 \\ \Rightarrow \quad & y\left(\frac{d y}{d x}\right)^2+2 x \frac{d y}{d x}-y=0 \end{aligned} $$

Hence, the correct answer is option: 'C'.