3. Given a point $P$ on the right branch of the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{3}=1$, the ratio of the distance from $P$ to the left focus $F_{1}$ to the distance from $P$ to the right focus $F_{2}$ is $2: 1$. Then $|O P|=$ $\qquad$
3. $\sqrt{39}$.

By the definition of a hyperbola, we know that $\left|P F_{1}\right|-\left|P F_{2}\right|=4$.
Combining this with $\left|P F_{1}\right|=2\left|P F_{2}\right|$, we get
$$
\left|P F_{2}\right|=4,\left|P F_{1}\right|=8 \text {. }
$$

Then, using the median length formula, we have
$$
\begin{array}{l}
|O P|^{2}=\frac{1}{2}\left(\left|P F_{1}\right|^{2}+\left|P F_{2}\right|^{2}\right)-\frac{1}{4}\left|F_{1} F_{2}\right|^{2} \\
=39 \\
\Rightarrow|O P|=\sqrt{39} .
\end{array}
$$