Kindly tell me which option is correct and what is reason for that
Why post here if you ignore the responses? Danny suggested you try x(t)= 0. That would give, of course, $\displaystyle a_k= 0$ for all k. And y(t)= 1 so $\displaystyle b_0= 1$ while $\displaystyle b_k= 0$ for all k> 0. That does not match "b".