​ If z1 is a complex number other than –1 such that |z1| = 1 and z2 = z1-1/z_1+1 then show that z2 is purely imaginary.

Question

If z₁ is a complex number other than –1 such that |z₁| = 1 and z₂ = \(\frac{z_1-1}{z_1+1}\) then show that z₂ is purely imaginary.

Answer 1

Let z₁ = a + ib such that | z₁| = √(a² + b²) = 1

Thus, the real part of z₂ is 0 and z₂ is purely imaginary.