A particle is moving in the xy -plane along a curve C passing...

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 $Q\left(x - \dfrac{y}{m}, 0\right)$

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

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

$m = \dfrac{y}{2 \cdot x} ⇒ 2\dfrac{dy}{y} = \dfrac{dx}{x}$

$2lnn = \ln x + lnk$

${y}^{2} = kx$

It passes through \left(3, 3\right) ⇒ k = 3$</p><p>curve$c ⇒ {y}^{2} = 3x$</p><p>Length of L.R.$= 3$</p><p>Focus$= \left(\dfrac{3}{4}, 0\right)$