Question

If O is the origin and Q is a variable point on y² = x, find the locus of the mid-point of OQ.

Answer 1

Let the coordinates of Q be (a, b), which lies on the parabola $y^2=x$.

$\Rightarrow b^2=a$ ... (1)

Let P(h, k) be the mid-point of OQ.

Now,

$$\begin{gathered} h=\frac{0+a}{2}\mathrm{and}k=\frac{0+b}{2} \\ \Rightarrow a=2h\mathrm{and}b=2k \end{gathered}$$

Putting a = 2h and b = 2k in equation (1), we get:

$$\begin{gathered} {\left(2k\right)}^2=2h \\ \Rightarrow 2k^2=h \end{gathered}$$

Hence, the locus of the mid-point of OQ is $2y^2=x$.