The definition of a hyperbola is
A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.
By this definition, the perpendicular line passing through the centre of the hyperbola, that is its conjugate axis should be called a hyperbola (may be a trivial hyperbola). As conjugate axis is set of all points such that the difference of their distance from from two foci is always $0$, which is always same, or remains constant. Is this a correct interpretation or am I missing something?