Question

$l x+m y=1$ is the equation of the chord $P Q$ of $y^{2}=4 x$ whose focus is $S$. If $P S$ and $Q S$ meet the parabola again at $R$ and $T$, respectively, then slope of $R T$ is

• A. $2 / m$
• B. $1 / m$
• C. $-1 / m$
• D. $-2 / m$

Answer 1

Answer: (C)

Let us take $P$ and $Q$ as $t_{1}$ and $t_{2}$, respectively.
Equation to $P Q$ is: $-2 x+y\left(t_{1}+t_{2}\right)=2 t_{1} t_{2}$
Comparing this with $l x+m y=1$, we get
$t_{1}+t_{2}=-2 m / l$ and $t_{1} t_{2}=-1 / l$
As $P S R$ and $Q S T$ are focal chords, coordinates of $R$ are $\left(1 / t_{1}{ }^{2},-2 / t_{1}\right)$ and that of $T$ are $\left(1 / t_{2}{ }^{2},-2 / t_{2}\right)$.
Slope of $R T=\frac{2\left(\frac{1}{t_{2}}-\frac{1}{t_{1}}\right)}{\frac{1}{\frac{1}{t_{1}^{2}}-\frac{1}{t_{2}^{2}}}}=-\frac{2 t_{1} t_{2}}{t_{1}+t_{2}}$
$=\frac{-2\left(\frac{-1}{l}\right)}{-\frac{2 m}{l}}=-\frac{1}{m}$