Let PQ be a focal chord of the parabola `y^(2)=4ax`. The tangents to the parabola at P and Q meet at point lying on the line
`y=2x+a,alt0`.
The length of chord PQ is

Options

A. 7a
B. 5a
C. 2a
D. 3a

Answer

B

Explanation

Let `(at_(1)^(2),2at_(1))" and "Q(at_(2)^(2),at_(2))` be the end-point of a focal chord of the parabola `y^(2)=4ax`. Then, `t_(1)t_(2)=-1" and "PQ=a(t_(2)-t_(1))^(2)`. The tangents at P and Q intersect at a point `(at_(1)t_(2),a(t_(1)+t_(2)))` which lies on y = 2x + a.
`:." "e(t_(1)+t_(2))=2at_(1)t_(2)+a`
`rArr" "t_(1)+t_(2)=-2+1+1`
`rArr" "t_(1)+t_(2)=-2+1=-1" "[becauset_(1)t_(2)=-1]`
`:." "PQ=a(t_(2)-t_(1))^(2)`
`=a{(t_(2)+ty_(1))^(2)-4t_(1)t_(2)}=a[(-1)^(2)-4(-1)]=5a`