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Tangent To a Parabola

If P1Q1 and...

Question

If $P_{1} Q_{1}$ and $P_{2} Q_{2}$ are two focal chords of the parabola $y^{2} = 4 a x$ then the chords $P_{1} P_{2}$ and $Q_{1} Q_{2}$ intersect on

A

tangent at the vertex of the parabola

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B

the directrix of the parabola

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C

At

x = 2 a

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D

y = 2 a

and

x = - 2 a

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Solution

The correct option is A the directrix of the parabola
$y^{2} = 4 a$
$P_{1} (a t_{1}^{2}, 2 a t_{1}), Q_{1} (a / t_{1}^{2}, - 2 a / t_{1})$
$P_{2} (a t_{2}^{2}, 2 a t_{2}), Q_{2} (a / t^{2^{2}}, - 2 a / t_{2})$
Chord $P_{1} P_{2}$
$y (t_{1} t_{2}) = 2 x + 2 a (t_{1} t_{2}) ∴ y (t_{1} t_{2}) - 2 x - 2 a (t_{1} t_{2}) = 0.... (1)$
Similarly $Q_{1} Q_{2}$
$y (\frac{- 1}{t} \frac{- 1}{t^{2}}) - 2 x - 2 a (\frac{- 1}{t_{1}} - \frac{1}{t_{2}}) = 0$
$∴ y (t_{1} t_{2}) + 2 x t_{1} t_{2} + 2 a = 0.... (2)$
$∴ (2) - (1) = 2 x (t_{1} t_{2} + 1) + 2 a (t_{2} t_{1}) = 0$
$= (x + a) (1 + t_{1} t_{2}) = 0$
$∴ x + a = 0$
$∴$ They intersect at the directriy

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