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Question

If a tangent to the parabola y2=8x meets the x-axis at T and intersect the tangent at vertex A at P, and the rectangle TAPQ is completed, then the locus of the point Q is

A
y2+2x=0
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B
x2+2y=0
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C
x22y=0
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D
y22x=0
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Solution

The correct option is A y2+2x=0

y2=8x
a=2
Vertex A is (0,0).
The tangent at any point (at2,2at) of the parabola y2=4ax is given by ty=x+at2.
Therefore, the tangent at any point of y2=8x is ty=x+2t2.
Since, the tangent at vertex A is y-axis,
so, T and P are (2t2,0) and (0,2t), respectively.

Let point Q be (h,k).
Then h=AT=2t2 (1)
and k=AP=2t (2)
Eliminating t, using (1) and (2), we get
2h=(k)2
k2+2h=0
Hence, the locus of Q is y2+2x=0

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