\[(a)\;{t_1}^2 =t_2t_3\quad (b)\;{t_2}^2=t_1t_3 \quad (c)\;{t_3}^2=t_1t_2 \quad (d)\;t_3=t_1+t_2\]

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Let 'u' be the velocity of swimmer in still water and 'v' be velocity of river.

In case I

Since v and u are $\perp$ to each other.

$t_1=2 \bigg(\large\frac{w}{\sqrt {u^2-v^2}}\bigg)$

In case II

the swimmer goes upstream and downstream

$t_2=\large\frac{w}{u+v}+\frac{w}{u-v}$

$\qquad=\large\frac{2uw}{u^2-v^2}$

In case III

Still water

$t_3=\large\frac{2w}{u}$

We see ${t_1}^2=t_2t_3$

Hence a is the correct answer.

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