Question

Let $PQ$ be a focal chord of $y^2=4ax$. The tangents to parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a(a>0)$. Length of $PQ$(in units) is :

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

Answer 1

The correct option is B $5a$

Let point of intersection of tangents at extremeties of focal chord be $R=(a{t}_1{t}_2,a(t_1+t_2\left)\right),t_1{t}_2=-1$
$R$ lies on the directix $x=-a$ and it also lies on line $y=2x+a$
$\therefore y=-a=a(t_1+t_2)$
$\Rightarrow t_1+t_2=-1$
$(t_2-t_1{)}^2=(t_1+t_2{)}^2-4t_1{t}_2=5$
$PQ=\sqrt{\left(a\right(t_1^2-t_2^2){)}^2+(2a(t_1-t_2){)}^2}$
$PQ=\sqrt{\left(a\right(t_1+t_2\left)\right(t_1-t_2){)}^2+(2a(t_1-t_2){)}^2}$
Putting values, we get $PQ=5a$