Suppose z is complex number with |z| = 1 (also assume that z is not 1 + 0*i)
Let z = a + b*i where a and b are real numbers.
Find a real parameter, t, such that
z = (t-i)/(t+i), where i = sqrt(-1)
|z| = sqrt(a^2+b^2)
The Attempt at a Solution
Since we are given |z| = 1, this implies that a^2 + b^2 = (a+b*i)*(a-b*i) = 1.
Also, (t-i)/(t+i) can be expressed as [1/(t^2+1)]*[(t^2-1)+(-2*t)*i].
Despite my efforts, I end up at two deadends: I reduce an equation to a tautology like 0 = 0 or I reduce to an expression for t that involves i (thereby making t imaginary instead of real).
Any help is greatly appreciated.