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If tangent at P and Q to the parabola y^...

If tangent at `P` and `Q` to the parabola `y^2 = 4ax` intersect at `R` then prove that mid point the parabola, where `M` is the mid point of `P` and `Q.`

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

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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