Question

Let O be the vertex and Q be any point on the parabola $x^2=8y$. If the point P divides the line segement OQ internally in the ratio 1 : 3, then the locus of P is

• A. $x^2=y$
• B. $y^2=x$
• C. $y^2=2x$
• D. $x^2=2y$

Answer 1

The correct option is D $x^2=2y$

Any point on the parabola $x^2=8y$ is $(4t,2t^2)$. Point P divides the line segment joining $O(0,0)$ and $Q(4t,2t^2)$ in the ratio 1 : 3. Apply the section formula for internal division.
Equation of parabola is
$x^2=8y$
let any point Q on this parabola is $(4t,2t^2)$.
Let P(h,k) be the point which divides the line segment joining (0,0) and $(4t,2t^2)$ in the ratio 1 : 3.

$\therefore h=\frac{1\times 4t+3\times 0}{4}\Rightarrow h=t\text{and}k=\frac{1\times 2t^2+3\times 0}{4}\Rightarrow k=\frac{t^2}{2}\Rightarrow k=\frac{1}{2}{h}^2\Rightarrow 2k=h^2[\because t=h]$
$\Rightarrow 2y=x^2,$ which is the required locus.