Question

Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,a>0$.
If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax$, then tan $\theta$ is equal to

• A. $\frac{2}{3}\sqrt{7}$
• B. $\frac{-2}{3}\sqrt{7}$
• C. $\frac{2}{3}\sqrt{5}$
• D. $\frac{-2}{3}\sqrt{5}$

Answer 1

The correct option is D $\frac{-2}{3}\sqrt{5}$


$m_{OP}=\frac{2at-0}{a{t}^2-0}=\frac{2}{t}$
$m_{OQ}=\frac{-2\frac{a}{t}}{\frac{a}{t^2}}=-2t\therefore \tan \theta =\frac{\frac{2}{t}+2t}{1-\frac{2}{t}.2t}=\frac{2(t+\frac{1}{t})}{1-4}$
where $t+\frac{1}{t}=\sqrt{5}$
$=\frac{2\sqrt{5}}{-3}$