Let C be a curve such that the normal at any point P on it meets x-axis and y-axis at A and B respectively.   If BP:PA=1:2 (internally) and the curve passes through the point (0,4) then which of the following alternative(s) is/are correct?

$\text{ The equation of normal of }\mathrm{P}(\mathrm{x},\mathrm{y})\text{ is }(\mathrm{Y}-\mathrm{y})=\frac{-1}{\frac{dy}{dx}}(\mathrm{X}-\mathrm{x})$

$\therefore A\left(x+y\frac{dy}{dx},0\right)\text{ and }B\left(0,y+\frac{x}{\frac{dy}{dx}}\right)$

$\mathrm{Now}\frac{1\left(x+y\frac{dy}{dx}\right)+2\left(0\right)}{1+2}=x\Rightarrow x+y\frac{dy}{dx}=3x$

$$\begin{gathered} \begin{array}{c}y\frac{dy}{dx}=2x\underset{\_}{\underset{\_}{\dots ..}\left(1\right)}\\ \Rightarrow \int ydy=\int 2xdx\Rightarrow \frac{y^2}{2}=x^2+C\end{array} \\ \text{ Also }(0,4)\text{ satisfy it, so }C=8 \\ \therefore {\mathrm{y}}^2=2{\mathrm{x}}^2+16\text{ (equation of curve) } \end{gathered}$$

$\text{ Which represent a hyperbola }$