7. For the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, the right focus is $F$. The line $l: y=k x+d$ does not pass through point $F$, and intersects the right branch of the hyperbola at points $P$ and $Q$. If the external angle bisector of $\angle P F Q$ intersects $l$ at point $A$, then the x-coordinate of point $A$ is $\qquad$
7. $\frac{a^{2}}{\sqrt{a^{2}+b^{2}}}$.

Draw a perpendicular line $h$ from point $A$ to the $x$-axis.
Let the projections of $P$ and $Q$ on line $h$ be $P^{\prime}$ and $Q^{\prime}$, respectively. Then $\frac{P P^{\prime}}{Q Q^{\prime}}=\frac{A P}{A Q}=\frac{F P}{F Q}$, which means $\frac{P F}{P P^{\prime}}=\frac{Q F}{Q Q^{\prime}}$. Therefore, line $h$ is the right directrix of the hyperbola. Hence, the x-coordinate of point $A$ is $\frac{a^{2}}{\sqrt{a^{2}+b^{2}}}$.