The length of the chord of contact of tangents drawn from $(x_{1}, y_{1})$ to the parabola $y^{2} = 4 a x$ is

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$\sqrt{(y_{1}^{2} - 4 a x_{1}) (y_{1}^{2} - 4 a^{2})} / a$

$\begin{aligned} \sqrt{(y_{1}^{2} - 4 a x_{1}) (y_{1}^{2} + 4 a^{2})} / a \end{aligned}$

$\begin{aligned} \sqrt{(y_{1}^{2} - 4 a x_{1}) / a} \end{aligned}$

$\sqrt{(y_{1}^{2} + 4 a x_{1}) (y_{1}^{2} - 4 a^{2})} / a$

answer is A.

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Detailed Solution

Let $P (a t_{1}^{2}, 2 a t_{1}), Q (a t_{2}^{3}, 2 a t_{2})$ be the points of contact of the tangents darwn from $A (x_{1}, y_{1})$ to the parabola $y^{2} = 4 a x .$

$∴ A =$ Point of intersection of the tangents at P and Q

$\Rightarrow (x_{1}, y_{1}) = (a t_{1} t_{2}, a [t_{1} + t_{2}]) \Rightarrow x_{1} = a t_{1} t_{2}, y_{1} = a (t_{1} + t_{2}) \Rightarrow t_{1} t_{2} = \frac{x_{1}}{a}, t_{1} + t_{2} = \frac{y_{1}}{a}$

Length of the chord of contact $t = P Q = \sqrt{(a t_{1}^{2} - a t_{2}^{2}) + {(2 a t_{1} - 2 a t_{2})}^{2}}$

$= a \sqrt{{(t_{1} + t_{2})}^{2} {(t_{1} - t_{2})}^{2} + 4 {(t_{1} - t_{2})}^{2}} = a \sqrt{[{(t_{1} + t_{2})}^{2} + 4] {(t_{1} - t_{2})}^{2}}$

$= a \sqrt{[{(t_{1} + t_{2})}^{2} + 4] [{(t_{1} + t_{2})}^{2} - 4 t_{1} t_{2}]} = a \sqrt{\frac{y_{1}^{2}}{a^{2}} + 4] [\frac{y_{1}^{2}}{a^{2}} - \frac{4 x_{1}}{a}]} = \frac{1}{a} \sqrt{(y_{1}^{2} + 4 a^{2}) (y_{1}^{2} - 4 a x_{1})}$