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 $\tan \theta =$

• 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 $\frac{-2}{3}\sqrt{5}$



As tangents drawn at end points of focal chord intersect at directrix
So, solving $y=2x+a,$ and $x=-a$ we get $(-a,-a)$
Equation of $PQ:(-a)y-2a(x-a)=0$
$\Rightarrow 2x+y-2a=0$
Solving it with parabola
$y^2-4ax\left(\frac{2x+y}{2a}\right)=0$
$\Rightarrow y^2-4x^2-2xy=0$
$\Rightarrow m^2-2m-4=0$
$\Rightarrow m_1+m_2=2,m_1{m}_2=-4$
$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$
$=\frac{\sqrt{(m_1+m_2{)}^2-4m_1{m}_2}}{1+m_1{m}_2}=-\frac{2\sqrt{5}}{3}$ (As angle is abtuse)