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Let a, r, s, t be non-zero real numbers....

Let a, r, s, t be non-zero real numbers. Let `P(at^(2),2at),Q(ar^(2),2ar)andS(as^(2),2as)` be distinct points on the parabola `y^(2)=4ax`. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K the point (2a,0).
If st=1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

A

`((t^(2) + 1)^(2))/(2t^(3))`

B

`(a(t^(2) + 1)^(2))/(2t^(3))`

C

`(a(t^(2) + 1)^(2))/(t^(3))`

D

`(a(t^(2) + 2)^(2))/(t^(3))`

Text Solution

Verified by Experts

The correct Answer is:
B
Plan Equation of tangent and normal at `(at^(2), 2at)` are given by `ty = x + at^(2)` and `y + tx = 2a + at^(3)`, respectively.
Tangent at `P + ty = x + at^(2)` or `y = (x)/(t) + at`
Normal at `S : y (x)/(t) = (2a)/(t) + (a)/(t^(3))`
Solving `2y = at + (2a)/(t) + (a)/(t^(3)) implies y = (a(t^(3) + 1)^(2))/(2t^(3))`
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