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Definition of Functions

Normals at th...

Question

Normals at three points P, Q, R to the parabola $y^{2} = 4 a x$ meet at the point A and S be its focus, if $| S P | . | S Q | . | S R | = λ (S A)^{2}$ then, $λ$ = ?

a^{3}

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a^{2}

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Solution

The correct option is A a
Normal : $x t + y = a t^{3} + 2 a t$
(a >0)
|PS| . |QS| . |RS|
$= a^{3} (t_{1}^{2} + 1) (t_{2}^{2} + 1) (t_{3}^{2} + 1)$
$= a^{3} [1 + (t_{1}^{2} + t_{2}^{2} + t_{3}^{2}) + {(t_{1} t_{2})^{2} + (t_{2} t_{3})^{2} + (t_{3} t_{1})^{2}} + (t_{1} t_{2} t_{3})^{2}]$
$= a [a^{2} + 2 a h - 4 a^{2} + 4 a^{2} + h^{2} - 4 a h + k^{2}]$
$= a [(h - a)^{2} + k^{2}]$
$= a (A S)^{2}$

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