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$\mathrm{PSQ}$ is a focal chord of a parabola whose focus is $\mathrm{S}$ and vertex is $\mathrm{A}$ . $\mathrm{PA}$ and $\mathrm{QA}$ are produced to meet the directrix in $\mathrm{R}$ and $\mathrm{T}$ respectively. Then $\mathrm{\angle RST}$ is equal to

Option: 1
$90^{\circ}$

Option: 2
$60^{\circ}$

Option: 3
$45^{\circ}$

Option: 4
$30^{\circ}$

Answers (1)

best_answer

Equation of A P is y= $\mathrm{\frac{2}{t_1} x}$
This meets directrix x=-a when

$\mathrm{y=-\frac{2 a}{t_1} }$
$\mathrm{\therefore \quad R_{\text {is }}\left(-a,-\frac{2 a}{t_1}\right) }$

Similarly T is $\mathrm{\left(-a,-\frac{2 a}{t_2}\right)}$ ; Slope of $\mathrm{R S=\frac{-2 a / t_1}{2 a}=-\frac{1}{t_1}}$
Slope of $\mathrm{T S=\frac{-2 a / t_2}{2 a}=-\frac{1}{t_2} }$

$\mathrm{\therefore}$ Product of slopes = $\mathrm{\left(-\frac{1}{t_1}\right)\left(-\frac{1}{t_2}\right)=\frac{1}{t_1 t_2}=\frac{1}{-1}=-1}\\ (\text{Q PQ is a focal chord} \Rightarrow t_1 t_2=-1 )$

$\mathrm{\therefore \quad R S \perp T S i.e. \angle R S T=90^{\circ}}$

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