\[(a)\;T\bigg [\frac{1}{4}-\frac{e}{2 \pi}\bigg] \quad (b)\;\frac{Te}{\pi} \quad (c)\;\bigg(\frac{e}{\pi}-1\bigg) T \quad (d)\;\frac{T \pi}{e}\]

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We know that areal velocity of planet is constant.

The desired time is time taken to travel from A to B

The time taken to cover complete area of ellipse =T

$\therefore \;t_{AB}=\bigg(\large\frac{Area\; of\; SAB}{Area\;of\;ellipse}\bigg)$$T$

Area of ellipse $=\pi ab$

Area of SAB $=\large\frac{1}{4} $$\pi a b-Area\;of \;\Delta\;OSB$

$\qquad=\large\frac{1}{4} $$\pi a b-\large\frac{1}{2}$$b \times ae$

$\therefore t_{AB}=\large\frac{\bigg(\Large\frac{\pi ab}{4}-\frac{1}{2} \large abe \bigg)T}{\pi a b}$

$\qquad=T\bigg [\large\frac{1}{4}-\frac{e}{2 \pi}\bigg]$

Hence a is the correct answer.

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