Let z_{1} and z_{2} be two complex numbers such that \left|\frac{z_{1}+z_{2}}{z_{1}-z_{2}}\right|=1, then what is Re\left(\frac{z_{1}}{z_{2}}\right)+1 equal to?

Explanation

The condition \left|\frac{z_1+z_2}{z_1-z_2}\right| = 1 implies |z_1+z_2| = |z_1-z_2|. This occurs if and only if z_1 and z_2 are perpendicular vectors in the complex plane. Thus, their quotient \frac{z_1}{z_2} is purely imaginary. Therefore, Re\left(\frac{z_1}{z_2}\right) = 0, and Re\left(\frac{z_1}{z_2}\right) + 1 = 0 + 1 = 1.

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