If two tangents from any point on the directrix to the parabola $y^{2} = 4 a x$ touch the parabola at $t_{1}, t_{2}$ then

t_{1} t_{2} = - 1

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t_{1} t_{2} = 1

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t_{1} t_{2} = 2

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t_{1} t_{2} = \frac{1}{2}

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Solution

The correct option is A $t_{1} t_{2} = - 1$
Pair of tangents from the directrix to the parabola are always perpendicular to each other.
$∴ m_{1} m_{2} = - 1$
$t_{1} y = x + a t_{1}^{2}$
$t_{2} y = x + a t_{2}^{2}$
$y_{1}^{1} = \frac{1}{t_{1}} = m_{1}$
$y_{1}^{1} = \frac{1}{t_{2}} = m_{2}$
$m_{1} m_{2} = \frac{1}{t_{1}} \times \frac{1}{t_{2}} = - 1$
$t_{1} t_{2} = - 1$

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If two tangents from any point on the directrix to the parabola $y^{2} = 4 a x$ touch the parabola at $t_{1}, t_{2}$ then