Question

If O is origin and R is a variable point on y² = 4x, then find the equation of the locus of the mid-point of segment OR

Answer 1

Let the variable point R be (x, y).

Let M(h, k) be the midpoint of R.

(h, k) = `((0 + x)/2, (0 + y)/2)`

= `(x/2, y/2)`

h  `x/2`

k = `y/2`

⇒ x = 2h, y = 2k

But R(x, y) is a point on y² = 4x

∴ (2k)² = 4(2h)

4k² = 8h

k² = 2h

The locus of M(h, k) is obtained by replacing h by x and k by y.

∴ The required locus is y² = 2x