If a tangent to the parabola $y^{2} = 4 a x$ meets the x-axis at T and the tangent at the vertex A, at P. Let the rectangle TAPQ be completed. Then the locus of the point Q is

a circle

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a straight line

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a parabola

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an ellipse

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Solution

The correct option is C
a parabola

The tangent at any point $B (a t^{2}, 2 a t)$ to the parabola is $t y = x + a t^{2} . . . . . . . (1)$

Since tangent at the vertex A is the y-axis, T and P are $(- a t^{2}, 0) a n d (0, a t)$ respectively. Clearly A is (0, 0)
If Q be(h, k), then h =AT = $- a t^{2}$ and k= AP = at Eliminating t, we get $k^{2} + a h = 0$
Hence the locus of Q is $y^{2} + a x = 0,$ which is parabola.

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A tangent to the parabola $y^{2} = 4 a x$ meets x axis at a point T. This tangent meets the tangent at the vertex at point P. If rectangle TAPQ is completed then the locus of Q is.

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y^{2} = 4 a x

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A tangent to the parabola $y^{2} = 4 a x$ meets x axis at a point T. This tangent meets the tangent at the vertex at point P. If rectangle TAPQ is completed then the locus of Q is.