Yes, this is a different property. In this curve, the particle has zero vertical acceleration, provided that it is started on the curve at the correct speed. In contrast, but brachistochrone is the curve connecting two given fixed points, which allows the particle to reach the lower point in the shortest possible time, if it is released from rest at the upper point. Similarly, the tautochrone is a curve for which a particle started from rest at any point on the curve will reach the bottom point in an equal time.

The brachistochrone and tautochrone are both actually the same curve, the cycloid, but the starting conditions are different. A cycloid is a brachistochrone if the upper point is on the cusp of the cycloid and the lower point is anywhere elso on it. It is a tautochrone if the lower point is at the bottom of the cycloid and the upper point is anywhere else on it.

Hope this helps!

--Stuart Anderson