If O is the origin and Q is a variable points on `x^2=4ydot` Find the locus of the mid pint of OQ.

Options

A. `y^(2) = 2x`
B. `y^(2) = x`
C. `x^(2) = y`
D. `x^(2) = 2y`

Answer

D

Explanation

Let the coordinates of Q be (a,b) and let P(h,k) be the mid-point of OQ. Then,
`h = (a+0)/(2) = (a)/(2) and k =(0+b)/(2) = (b)/(2) rArr a = 2h and b = 2k ……(i)`
Here, a and b are two variable which are to be eliminated.
Since, (a,b) lies on `x^(2) = 4y`
`therefore " "a^(2) = 4b`
`rArr (2h)^(2) = 4 (2k)" "["using (i)"]`
`rArr h^(2) = 2k`
Hence, the locus of (h,k) is `x^(2)= 2y` .