The maximum distance from origin to the point z satisfying the equation $\big|z+\large\frac{1}{z}\big|=a$ is

Question

$\begin{array}{1 1}(A)\;\large\frac{1}{2}\normalsize (\sqrt{a^2+1}+a)&(B)\;\large\frac{1}{2}\normalsize (\sqrt{a^2+2}+a)\\(C)\;\large\frac{1}{2}\normalsize (\sqrt{a^2+4}+a)&(D)\;\text{None of these}\end{array} $