Let O be the vertex and Q be any point on the parabola x^2=8y.

Question

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

Accepted Answer

Let the coordinates of Q and P be (x1, y1) and (h, k) respectively.

$$\because$$ Q lies on x2 = 8y,

$$\therefore$$ x$_1^2$ = 8y ....... (1)

Again, P divides OQ internally in the ratio 1 : 3.

$$\therefore$$ $$h = {{{x_1} + 0} \over 4} = {{{x_1}} \over 4}$$ or x1 = 4h and

$$k = {{{y_1} + 0} \over 4} = {{{y_1}} \over 4}$$ or y1 = 4k

Now putting x1 and y1 in (1) we get,

16h2 = 32k or, h2 = 2k

$$\therefore$$ the locus of P is given by, x2 = 2y.

Solutions · 1