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A tangent is drawn to the parabola y^2=4...

A tangent is drawn to the parabola `y^2=4a x` at `P` such that it cuts the y-axis at `Qdot` A line perpendicular to this tangents is drawn through `Q` which cuts the axis of the parabola at `R` . If the rectangle `P Q R S` is completed, then find the locus of `Sdot`

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The correct Answer is:
`2a(x-a)-y^(2)=(a-x)(a((x-a)/(y))^(2)-x)`
From the property of tangent of parabola, R is focus.

From the figure, product of slope of SR and PS is -1.
`:." "(2at-k)/(at^(2)-h)xx(k-0)/(h-a)=-1` (1)
Slope of tangent at PQ = Slope of RS
`:." "(k)/(h-a)=(1)/(t)rArrt=(h-a)/(k)`
Putting the value of t into (1), we get
`(2a((h-a)/(k))-k)k=(a-h)(a((h-a)/(k))^(2)-h)`
`rArr" "2a(x-a)-y^(2)=(a-x)(a((x-a)/(y))^(2)-x)`,
which is the required locus.
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