I have a 20 amp dedicated electrical line powering a motor which draws 9.7 amps. I want to put a second one in as a backup. What is the probability that the 20 amp circuit breaker would trip if both started at the same time?

Without more specific information about the motor, I can’t provide the exact answer you need. But generally speaking, there is a good possibility that it would probably trip, as motors draw multiple times their rated amperage during initial startup.