Question

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

• A. $9x^2-12y=8$
• B. $4x^2-3y=2$
• C. $x^2-3y=2$
• D. $9x^2-3y=2$

Answer 1

The correct option is A $9x^2-12y=8$

Let point $P$ be $(2t,t^2)$ and $Q$ be $(h,k)$.
By section formula,
$h=\frac{2t}{3},k=\frac{-2+t^2}{3}$
Now, eliminating $t$ from the above equations, we get
$3k+2={\left(\frac{3h}{2}\right)}^2$
Replacing $h$ and $k$ by $x$ and $y$, we get the locus of the curve as $9x^2-12y=8$