Let a, r, s, t be non-zero real numbers. Let `P(at^2, 2at), Q, R(ar^2, 2ar) and S(as^2, 2as)` be distinct points onthe parabola `y^2 = 4ax`. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K isthe point (2a, 0). The value of r is

Options

A. `-(1)/(t)`
B. `(t^(2)+1)/(t)`
C. `(1)/(t)`
D. `(t^(2)-1)/(t)`

Answer

D

Explanation

4 Slope (PK)=Slope (QR)
`(2at-0)/(at^(2)-2a)=((-2a)/(t)-2ar)/((a)/(t^(2))-ar^(2))`
`rArr(t)/(t^(2)-2)=-(((1)/(t)+r)/((1)/(t^(2))-r^(2)))`
`rArr(1)/(t^(2)-2)=(1+rt)/(t^(2)r^(2)-1)`
`rArrt^(2)r^(2)-1=t^(2)+rt^(3)-2-2rt`
`rArrt^(2)r^(2)+(2-t^(2))tr+(1-t^(2))=0`
`rArrtr=-1ortr=t^(2)-1`
`rArrr=-1//t or r=(t^(2)-1)/(t)`
But for `r=-1//t`, points Q and R are coincident.
`:." "r=bar(" ")`