Thanks very much for your response and effort, Ackbach. Good to know that simple intuition still has value!
I must assume that whoever found the correct value (0.551915024494) did so using a more clever route than I chose. Although I did find a method that gave a 96% result (but not a 100%!).
I am trying to determine the optimum length of the control vectors for a cubic Bézier approximation of a $90^\circ$ circular arc. However to my limited mathematical mind the equation cannot be solved algebraically. Wolfram Alpha even finds it too complicated. However I am told that it IS...