Question

$lx+my=1$ is the equation of the chord $PQ$ of $y^2=4x$ whose focus is $S$. If $PS$ and $QS$ meet the parabola again at $R$ and $T$, respectively, then slope of $RT$ 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 $PQ$ is: $-2x+y\left(t_1+t_2\right)=2t_1{t}_2$
Comparing this with $lx+my=1$, we get
$t_1+t_2=-2m/l$ and $t_1{t}_2=-1/l$
As $PSR$ and $QST$ 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 $RT=\frac{2\left(\frac{1}{t_2}-\frac{1}{t_1}\right)}{\frac{1}{\frac{1}{t_1^2}-\frac{1}{t_2^2}}}=-\frac{2t_1{t}_2}{t_1+t_2}$
$=\frac{-2\left(\frac{-1}{l}\right)}{-\frac{2m}{l}}=-\frac{1}{m}$