Example 6. Find the longest common chord passing through one of the intersection points of two intersecting circles.
Solve As shown in Figure 6, connect $\mathrm{O}_{1} \mathrm{O}_{2}$, draw $\mathrm{AB} / / \mathrm{O}_{1} \mathrm{O}_{2}$ through $Q$, draw any common chord $\mathrm{GD}$ through the foot of $Q$, draw $\mathrm{O}_{\text {\& }} \mathrm{E} \perp \mathrm{GD}$, $\mathrm{O}_{2} \mathrm{~F} \perp \mathrm{CD}$, $\mathrm{O}_{2} \mathrm{~N} \perp \mathrm{O}_{1} \mathrm{E}$, in the right triangle $\Delta \mathrm{O}_{1} \mathrm{NO}_{2}$, since $\mathrm{O}_{1} \mathrm{O}_{2}>\mathrm{O}_{2} \mathrm{~N}$, $\mathrm{CD}=2 \mathrm{O}_{2} \mathrm{~N}$, $\mathrm{AB}=2 \mathrm{O}_{1} \mathrm{O}_{2}$, thus $\mathrm{AB}>\mathrm{CD}$. However, due to the arbitrariness of $\mathrm{CD}$, it is known that $\mathrm{AB}$ is the longest chord among all common chords passing through point $Q$.