The question is: Prove that if the velocity and acceleration vectors of a parameterized curve X(t) are always perpendicular, the speed equals a constant. 

My attempt: I figured I could let the curve be given by a function that I know to always have perpendicular velocity and acceleration vectors, which is the equation of a circle. Then I made the parameters be x(t) = cos(t) and y(t) = sin(t), and since speed is the square root of the velocity vector dotted with itself, it is easy to see here that the speed would always be 1, a constant. 

That is the only way I can figure out how to arrive at an answer, but it doesn't feel right to me. I feel like I'm not being general enough, or that I'm not being rigorous in my supposition that X(t) is necessarily given by a circle. Was this a valid way of doing this proof?