If the point t is one extremity of a focal chord of the parabola $y^{2} = 4 a x$ then the length of the chord is

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Solution

The correct option is D
If $t_{1}$ is the other end of the focal chord, then the equation of the chord is $y (t + t_{1}) = 2 x + 2 a t t_{1}$
Since it passes through the focus (a, 0), we get $2 a + 2 a t t_{1} = 0 \Rightarrow t t_{1} = - 1 \Rightarrow t_{1} = - \frac{1}{t}$
Length of the chord = $\sqrt{(a t^{2} - a t_{1}^{2})^{2} + (2 a t - 2 a t_{1})^{2}} = a \sqrt{(t^{2} - \frac{1}{t^{2}})^{2} + 4 (t + \frac{1}{t})^{2}}$
$= a (t + \frac{1}{t}) \sqrt{(t - \frac{1}{t})^{2} + 4} = a (t + \frac{1}{t})^{2}$

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