The differential equation satisfied by the system of parabolas \({y}^{2}=4a(x+a)\) is: [JEE Main 2021, 18 Mar (Shift 1)]

\(\mathrm{y}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}-2\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{y}=0\)

\(\mathrm{y}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}-2\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)+\mathrm{y}=0\)

\(\mathrm{y}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}+2\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{y}=0\)

\(\mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)+2\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)-\mathrm{y}=0\)