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Tangent and normal drawn to a parabola a...

Tangent and normal drawn to a parabola at `A(a t^2,2a t),t!=0` meet the x-axis at point `B` and `D` , respectively. If the rectangle `A B C D` is completed, then the locus of `C` is

A

y=2a

B

`x=2a-(y^(2))/(4a)`

C

x=2a

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
(2) The equation of tangent and normal at A are `yt=x+at^(2)andy=-tx+2at+at^(3)`, respectively.
Therefore, `B-=(-at^(2),0)andD-=(2a+at^(2),0)`. If ABCD is a rectangle, then the midpoint of BD and AC will be coincident. So,
`h+at^(2)=2a+at^(2)-at^(2)`,
k+2at=0
`orh=2a-at^(2),t=-(k)/(2a)`
Therefore, the locus is
`x=2a-(y^(2))/(4a)`
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