5. Let the line $x=2$ intersect the asymptotes of the hyperbola $\Gamma: \frac{x^{2}}{4}-y^{2}=1$ at points $E_{1}$ and $E_{2}$, and let $\overrightarrow{O E_{1}}=e_{1}, \overrightarrow{O E_{2}}=e_{2}$. For any point $P$ on the hyperbola $\Gamma$, if $\overrightarrow{O P}=a e_{1}+b e_{2}$ $(a, b \in \mathbf{R})$, then $(\quad)$.
(A) $0<a^{2}+b^{2}<1$
(B) $0<a^{2}+b^{2}<\frac{1}{2}$
(C) $a^{2}+b^{2} \geqslant 1$
(D) $a^{2}+b^{2} \geqslant \frac{1}{2}$
5. D.

It is easy to find $E_{1}(2,1), E_{2}(2,-1)$. Then $\overrightarrow{O P}=a e_{1}+b e_{2}=(2 a+2 b, a-b)$. Since point $P$ is on the hyperbola, we have
$$
\frac{(2 a+2 b)^{2}}{4}-(a-b)^{2}=1 \text {. }
$$

Simplifying, we get $4 a b=1$.
Therefore, $a^{2}+b^{2} \geqslant 2 a b=\frac{1}{2}$.