Question

The symmetry of a hyperbola with a center at (h, k) only occurs at y = k. True or False

Answer 1

A hyperbola with a center at (0, 0) can be defined  as x²/a² − y²/b² = ±1. 
The statement "The symmetry of a hyperbola with a center at (h, k) only occurs at y = k" is false, because a hyperbola have many different orientations.
It doesn't have to be symmetric about the lines y = k or x = h.