A particle is moving in the x y -plane along a curve C passing through the point 3 , 3 . The tangent to the…

Let Point $P\left(x,y\right)$

So equation of line passing through $P\left(x,y\right)$ will be

 $Y-y=m\left(X-x\right)$

Point on $x$-axis is given by $\mathrm{Q}\left(\mathrm{x}-\frac{\mathrm{y}}{\mathrm{m}},0\right)$

Mid-point of $PQ$ lies on $y$ axis so its $x$ co-ordinate will be zero,

$x-\frac{y}{m}+x=0$

$m=\frac{y}{2·x}\Rightarrow 2\frac{dy}{y}=\frac{dx}{x}$

$2\mathrm{lnn}=\ln x+\ln k$

$y^2=kx$

It passes through $\left(3,3\right)\Rightarrow k=3$

curve $c\Rightarrow y^2=3x$

Length of L.R. $=3$

Focus$=\left(\frac{3}{4},0\right)$