A normal drawn at a point P on the parabola `y^2 = 4ax` meets the curve again at Q. The least distance of Q from the axis of the parabola, is

Options

A. `4sqrt6a`
B. `2sqrt6a`
C. `3sqrt6a`
D. none of these

Answer

A

Explanation

Let `P(at^(2), 2at)` be a point on the parabola `y^(2)=4ax` such that it culs the parabola again at `Q(at_(1)^(2), 2at_(1))`.
Then,
`t_(1)=-t-2/t`
`rArr" "|t_(1)|=|t+2/t|`
`rArr|t_(1)|ge|2sqrt(txx2/t)|" "["Using :"AMgeGM]`
`rArr" "|t_(1)|ge2sqrt2`
Now,
`OQ=sqrt((at_(1)^(2)-0)^(2)+(2at_(1)-0)^(2))`
`rArr" "OQ=at_(1)sqrt(t_(1)^(2)+4)`
`rArr" "OQge2asqrt2sqrt(8+4)rArrOQge4sqrt6a`
Clearly, minimum value of OQ is `(sqrt6xx"Latusrectum ")`