The normal to a curve at P(x,y) meets the x-axis at G. If the distance of G from the origin is three times the abscissa of P, then the curve is a

Slope of normal $=-\frac{\mathrm{dx}}{\mathrm{dy}}$

This equation of normal is $\mathrm{Y}-\mathrm{y}=-\frac{\mathrm{dx}}{\mathrm{dy}}(\mathrm{X}-\mathrm{x})$
$\mathrm{G}=\left(\mathrm{x}+\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}},0\right);\mathrm{OG}=3\mathrm{x}\Rightarrow \mathrm{x}+\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}=3\mathrm{x}$

Integrating we get $\frac{{\mathrm{y}}^2}{2}=\frac{2{\mathrm{x}}^2}{2}+\mathrm{c}$
$\Rightarrow \frac{{\mathrm{y}}^2}{2}-{\mathrm{x}}^2=\mathrm{c}$ ,which is hyperbola