Question

If the distance of two points $P$ and $Q$ on the parabola $y^{2}=4 a x$ from the focus of a parabola are $4$ and $9$ respectively then the distance of the point of intersection of tangents at $P$ and $Q$ from the focus is

• A. $8$
• B. $6$
• C. $3.5$
• D. $13$

Answer 1

Answer: (B)

Let $T$ is the point of the intersection of the tangents at $P, Q$.
We have $S P=a+a t_{1}^{2}, S Q=a+a t_{2}^{2}$
Also $T \equiv\left(a t_{1} t_{2}, a\left(t_{1}+t_{2}\right)\right)$
Now $S T^{2}=\left(a-a t_{1} t_{2}\right)^{2}+\left(a\left(t_{1}+t_{2}\right)\right)^{2}$
$=a^{2}\left(1+t_{1}{ }^{2} t_{2}{ }^{2}+t_{1}{ }^{2}+t_{2}{ }^{2}\right)$
$=a^{2}\left(1+t_{1}^{2}\right)\left(1+t_{2}^{2}\right)=\left(a+a t_{1}^{2}\right)\left(a+a t_{2}{ }^{2}\right)$
$=S P \times S Q=4 \times 9=36$
$\therefore S T=6$