Length of the chord of contact of the tangents drawn from (x1, y1) to the parabola y2=4ax is

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

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

$\frac{\sqrt{{(y_{1}^{2} - 4 a x_{1})}^{3}}}{2 a}$

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

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

answer is A.

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Detailed Solution

Suppose that the ends of chord of contact are $P (t_{1}), Q (t_{2})$

The point of intersection of tangents drawn at the above points to the given parabola is $(a t_{1} t_{2}, a (t_{1} + t_{2})) = (x_{1}, y_{1})$

The length of the chord of contact is distance between the points $P$ and $Q$

$\begin{array}{rcl} d & = & \sqrt{a^{2} {({t_{2}}^{2} - {t_{1}}^{2})}^{2} + 4 a^{2} {(t_{2} - t_{1})}^{2}} \\ = & a \sqrt{{(t_{2} - t_{1})}^{2} ({(t_{2} + t_{1})}^{2} + 4)} \\ = & a \sqrt{[{(t_{2} + t_{1})}^{2} - 4 t_{1} t_{2}] [({(t_{2} + t_{1})}^{2} + 4)]} \\ = & \frac{\sqrt{(y^{} - 4 a x) (y^{2} - 4 a^{2})}}{|a|} \end{array}$