If $P$ is a point which divides the line segment joining the focus of the parabola $y^2=12 x$ and a point on…
If $P$ is a point which divides the line segment joining the focus of the parabola $y^2=12 x$ and a point on the parabola in the ratio 1:2. Then the locus of $P$ is
$y^2=2(x-2)$
$y^2=4 x$
$y^2=4(x-2)$
$y^2=9(x-3)$
Solution
Parabola : $y^2=12 x \Rightarrow \quad$ Focus $=(3,0)$
Let $Q\left(3 t^2, 6 t\right)$ be a point on the parabola.
$\begin{aligned}
& \Rightarrow P(x, y)=\left(\frac{3 t^2+6}{3}, \frac{6 t}{3}\right)=\left(t^2+2,2 t\right) \\
& \Rightarrow x=t^2+2, y=2 \mathrm{t} \Rightarrow t=\frac{y}{2} \\
& \therefore x=\frac{y^2}{4}+2 \Rightarrow y^2=4(x-2)
\end{aligned}$