Question

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

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

Answer 1

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

Any point on the parabola $x^2=8y$ is $(4t,2t^2).$

Point $P$ divides the line segment joining of $O(0,0)$ and $Q(4t,2t^2)$ in the ratio $1:3$. Apply the section formula for the internal division.

The equation of parabola is $x^2=8y$

Let any $Q$ on the 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 $1:3$

$\Rightarrow$ $h=\frac{1\times 4t+3\times 0}{4}$

$\Rightarrow$ $h=t$

$\Rightarrow$ $k=\frac{1\times 2t^2+3\times 0}{4}$

$\Rightarrow$ $k=\frac{t^2}{2}$

$\therefore$ $k=\frac{1}{2}{h}^2$

$\Rightarrow$ $2k=h^2$

$\Rightarrow$ $x^2=2y$