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Question

Let r be a real number, |r| < 2, and let z be a complex number such that z + 1z = r.Find |z|.

Accepted Answer

From the equation z + 1z = r, z^2 + 1 = rz, so z^2 - rz + 1 = 0.By the quadratic equation, z = r r^2 - 42 = r i 4 - r^22.Then |z| = ( r2 )^2 + ( 4 - r^22 )^2 = r^24 + 4 - r^24 = 1.