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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).
The value of r is

A

`=(1)/(t)`

B

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

C

`(1)/(t)`

D

`(t^(2) - 1)/(t)`

Text Solution

Verified by Experts

The correct Answer is:
D
Plan (I) If `P(at^(2), 1at)` is one end point of focal chord of parabola `y^(2) = 4ax`, then other end point is `((a)/(t^(2)), (2a)/(t))`
(ii) Slope of line joining two points `(x_(1), y_(1))` and `(x_(2), y_(2))` is given by `(y_(2) - y_(1))/(x_(2) - x_(1))`
If PQ is focal chord, then coordinates of Q will be
`((a)/(t^(2)),(2a)/(t))`
Now, slope of QR = slope of PK
`(2ar + (2a)/(t))/(ar^(2) - (a)/(t^(2))) = (2 at)/(at^(2) - 2a) implies (r + 1//t)/(r^(2) - 1//t^(2)) = (t)/(t^(2) - 2)`
`implies (1)/(r - (1)/(t)) = (t)/(t^(2) - 2) implies r - (1)/(t) = (t^(2) - 2)/(t) = t - (2)/(t)`
`implies r = t - (1)/(t) = (t^(2) - 1)/(t)`
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