In theory, you'd only be able to build such a cable at or near the equator, for that reason. Some suggest anchoring it with an asteroid or something in geostationary orbit or slightly above to keep it taught without having to extend it out super far.
Discussion
The center of gravity of a space elevator has to be slightly above GEO and designs typically extend very far out as that's the most efficient way to get tons of lift from little material **and** you can keep climbing that cable beyond GEO to get extra speed from that gigantic whip.