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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
It is given thatPQ is a focal chord of the parabola `y^(2)=4ax,` So, the coordinates of Q are `(a/t^(2), (-2a)/t)`

Now,
Slope of QR=Slope of PK
`((-2a)/t-2ar)/(a/t^(2)-ar^(2))=(2at-0)/(at^(2)-2a)`
`rArr" "-((1/t+r)/(1/t^(2)-t^(2)))=t/(t^(2)-2)rArr1/(r-1/t)=t/(r-1/t)=t/(t^(2)-2)rArrr=(t^(2)-1)/t`
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