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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
Solution
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
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