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`.
If chord PQ subtends an angle `theta` at the vertex of `y^(2)=4ax`, then `tantheta=`

Options

A. `(2sqrt7)/3`
B. `-(2sqrt7)/3`
C. `(2sqrt3)/3`
D. `-(2sqrt5)/3`

Answer

D

Explanation

Let `P(at_(1)^(2), 2at_(1))" and "(Q(at_(2)^(2), 2at_(2))` be the end-points of a focal chord of the parabola `y^(2)=4ax,` Then, `t_(1)t_(2)=-1`. The tangents at P and Q intersect at a point which lies on the line `y=2x+a`.
`:." "a(t_(1)+t_(2))=2at_(1)t_(2)+a`
`rArr" "t_(1)+t_(2)=-2+1=-1" "[becauset_(1)t_(2)=-1]`
Now, `(t_(1)-t_(2))^(2)=(t_(2)+t_(1))^(2)-4t_(1)t_(2)`
`rArr" "(t_(2)-t_(1))^(2)=1+4=5`
`rArr" "(t_(2)-t_(1))=+-sqrt5`
Let O be the vertex of the parabola `y^(2)=4ax`. Then, slopes of OP and OQ are `2/t_(1)" and "2/t_(2)` respectively. It is given that PQ subtands an angle `theta` at the vertex O of the parabola.
`:." "tantheta=(2/t_(1)-2/t_(2))/(1+2/t_(1)xx2/t_(2))=(2(t_(2)-t_(1)))/(t_(1)t_(2)+4)=(2(t_(2)-t_(1)))/(-1+4)=2/3(t_(2)-t_(1))`
`rArr" "tantheta=+-(2sqrt5)/3`
Clearly, `theta` is obtuse. Therefore, `tan theta =-(2sqrt5)/3`.