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,
\[h = \frac{0 + a}{2}\text{ and }k = \frac{0 + b}{2}\]
\[ \Rightarrow a = 2h\text{ and }b = 2k\]
Putting a = 2h and b = 2k in equation (1), we get:
\[\left( 2k \right)^2 = 2h\]
\[ \Rightarrow 2 k^2 = h\]
Hence, the locus of the mid-point of OQ is \[2 y^2 = x\]