The differential equation satisfied by the system of parabolas $\mathrm{y}^{2}=4 \mathrm{a}(\mathrm{x}+\mathrm{a})$ is :
Correct Option: , 3
$y^{2}=4 a x+4 a^{2}$
differentiate with respect to $x$
$\Rightarrow 2 \mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}=4 \mathrm{a}$
$\Rightarrow a=\left(\frac{y}{2} \frac{d y}{d x}\right)$
so, required differential equation is
$\mathrm{y}^{2}=\left(4 \times \frac{\mathrm{y}}{2} \frac{\mathrm{dy}}{\mathrm{dx}}\right) \mathrm{x}+4\left(\frac{\mathrm{y}}{2} \frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}$
$\Rightarrow \mathrm{y}^{2}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}+2 \mathrm{xy}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{y}^{2}=0$
$\Rightarrow \mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}+2 \mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{y}=0$
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