Explanation

(d)
In $\mathrm{\Delta}\ OCA$, tan $30{^\circ} = \frac{AC}{OC}$
$\Rightarrow \ \frac{1}{\sqrt{3}} = \frac{2at}{{at}^{2}} \Rightarrow t = 2\sqrt{3}$
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Again , in $\mathrm{\Delta}\ OCA$
$OA = \sqrt{{OC}^{2} + {AC}^{2}}$
$= \sqrt{{({at}^{2})}^{2} + {(2at)}^{2}}$
$= \sqrt{\lbrack\left( 2{\sqrt{3)}\ }^{2} \right\rbrack^{2}a^{2} + {4a}^{2}\left( 2\sqrt{3} \right)^{2}}$
$= \sqrt{{144a}^{2} + {48a}^{2}} = \sqrt{{192a}^{2}}$
$\Rightarrow OA = 8\sqrt{3}a$