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Q.

If the normals at three points, P,Q,R of the parabola $y^{2} = 4 a x$ meet in a point O and S be its focus, then $|S P| . |S Q| . |S R|$ is equal to

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a

$a^{2}$

b

$a^{2} {(S O)}^{3}$

c

$a {(S O)}^{2}$

d

$a {(S O)}^{3}$

answer is C.

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Detailed Solution

$S (a, 0) p (a t_{1}^{2}, 2 a t_{1}) Q (a t_{2}^{2}, 2 a t_{2}), R (a t_{3}^{2}, 2 a t_{3})$

$O (h, k) = (2 a - a (t_{1} t_{2} + t_{2} t_{3} + t_{3} t_{1}), a t_{1} t_{2} t_{3})$

$|S P| |S Q| |S R| = (a + a t_{1}^{2}) (a + a t_{2}^{2}) (a + a t_{3}^{2}) = a {(S O)}^{2}$

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