$\begin{array}{1 1}(A)\;4a \;cosec ^2 \theta \\(B)\;4a\; \sec^2 \theta \\(C)\;a \;cosec^2 \theta \\(D)\;none\; of\; these \end{array}$

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Length of the focal chord if

$a \sqrt {(t^2 -\large\frac{1}{t^2})^2 +4 (t+\frac{1}{t})^2}$

$\qquad= a \bigg( t+\large\frac{1}{t} \bigg)^2$

Now $\tan \theta =\large\frac{2( t +\large\frac{1}{t})}{t^2-\frac{1}{t^2}}$

$\qquad= \large\frac{2}{t-\large\frac{1}{t^2}}$

=> $t- \large\frac{1}{t}$$= 2 \cot \theta$

=> $ \bigg(t +\large\frac{1}{t}\bigg)^2 =\bigg(t -\large\frac{1}{t} \bigg)^2 +4$

$\qquad= 4 \cot ^2 \theta+4$

$\qquad= 4 a \;cosec ^2 \theta$

Hence A is the correct answer.

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