Why do astronomers believe that the engine at the center of a quasar is a supermassive black hole?
Quasars are small and emit such huge amounts of energy that a supermassive black hole is the best known explanation of their power source.
Quasars emit vast amounts of energy for long periods of time. A supernova explosion can emit huge amounts of energy but only for a few weeks.
Quasars energy output changes with a period of days or months. This means that the source of the energy must be quite small - in the order of the size of our solar system.
Supermassive black holes have been observed at the centres of many galaxies including our own. It is know thought that every galaxy has a supermassive black hole at its centre which drives the evolution of the galaxy.
A supermassive black hole can have material falling into it in what is know as an accretion disc. If plenty of additional material falls into the accretion disc it gets super heated by friction and gravitational effects to the point where it emits huge amounts of energy. This is a quasar.