The locus of a point which divides the line segment
Question:

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $x^{2}=4 y$, internally in the ratio $1: 2$, is:

1. (1) $9 x^{2}-12 y=8$

2. (2) $9 x^{2}-3 y=2$

3. (3) $x^{2}-3 y=2$

4. (4) $4 x^{2}-3 y=2$

Correct Option: 1

Solution:

Let point $P$ be $\left(2 t, t^{2}\right)$ and $Q$ be $(h, k)$

Using section formula,

$h=\frac{2 t}{3}, k=\frac{-2+t^{2}}{3}$

Hence, locus is $3 k+2=\left(\frac{3 h}{2}\right)^{2}$

$\Rightarrow \quad 9 x^{2}=12 y+8$