no it doesn't.
one way to think of it is that the tropic of capricorn is the furthest south the sun's path goes, and the tropic of cancer is the northern most line that the sun traces out.
since neither the tropic of capriforn nor the tropic of cancer are on the equator, the sun must be tracing out a different line and hence be on a different plane.
also, if the sun was on the same plane as the equator, then we wouldn't have seasons.
