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$\mathrm{P S Q}$ is a focal chord of a parabola whose focus is S and vertex is A. P is joined to vertex A and extended to meet the directrix at R. Q is also joined to A and extended to meet the directrix at T. Then, $\mathrm{\angle R S T}$ is equal to
Option: 1
45

Option: 2
60

Option: 3
90

Option: 4
30

Answers (1)

best_answer

Let the parabola be $\mathrm{y^2=4 a x}$ . Let $\mathrm{P, Q}$ be $\mathrm{\left(a t_1^2, 2 a t_1\right)}$ and $\mathrm{\left(a t_2{ }^2, 2 a t_2\right)}$ respectively.

As $\mathrm{P S Q}$ is a focal chord $\mathrm{\Rightarrow t_1 t_2=-1}$ $.......(1)$

The equation of the directrix is $\mathrm{x+a=0.}$

Equation of $\mathrm{P A}$ is $\mathrm{y=\frac{2 a t_1}{a t_1^2} x \Rightarrow y=\frac{2}{t_1} x}$

$\mathrm{\therefore \quad Point \, \, R \, \, is \left(-a, \frac{-2 a}{t_1}\right)}$
Similarly, point T is $\mathrm{\left(-a, \frac{-2 a}{t_2}\right)}$

S is (a, 0)

$\mathrm{\therefore \quad}$ (Slope of ST)( Slope of RS)

$\mathrm{=\frac{2 a \backslash t_2}{2 a} \cdot \frac{2 a / t_1}{2 a}=\frac{1}{t_1 t_2}}$

$\mathrm{=-1 \quad \, \, \, \, \, from (1)}$
Hence, $\mathrm{\angle T S R=90^{\circ}.}$

The answer is (c)

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