A particle is moving in the xy-plane along a curve C passing through the point (3,3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with

$$\begin{gathered} Tangent at P(x, y) is \\ \mathrm{Y}-\mathrm{y}=\mathrm{m}(\mathrm{X}-\mathrm{x}) \\ point at x-axis is Q(x - \frac{y}{m}, 0) \\ point at y axis is R(0, y - mx) \\ Now R is mid point at PQ \\ \therefore O=\frac{x+x-\frac{y}{m}}{2} \\ \Rightarrow 2x=\frac{y}{m}\Rightarrow \frac{dy}{dx}=\frac{y}{2x} \\ \Rightarrow \frac{2}{y}dy=\frac{1}{x}dx \\ \Rightarrow 2\ln y=\ln x+lnc \\ \Rightarrow y^2=cx \\ it passes through (3,3) \\ \Rightarrow 9 = 3c \Rightarrow c = 3 \\ \therefore y^2=3x \end{gathered}$$