PSQ is a focal chord of a parabola whose focus is S and vertex is A. PA and QA are produced to meet the directrix in R and T, respectively. Then `angleRST=``

Options

A. `90^(@)`
B. `60^(@)`
C. `45^(@)`
D. `30^(@)`

Answer

A

Explanation

Let `P(at_(1)^(2), 2at_(1))" and "Q(at_(2)^(2), 2at_(2))` be the end-points of focal chord PQ of the parabola `y^(2)=4ax`. Then, `t_(1)t_(2)=1`
Equations of AP are `y=2/t_(1)x" and "y=2/t_(2)x` respectively.
These two intersect `x= -a` and T whose coordinates are `(-a, -(2a)/t_(1))" and "(a, (2a)/t_(1))` respectively.

`:." "m_(1)="Slope of RS"=1/t_(1), m_(2)="Slope of TS"=1/t_(2)`
`:." "m_(1)m_(2)=1/(t_(1)t_(2))=-1" "[becauset_(1)t_(2)=-1]`
`rArr" "/_RST=90^(@)`