Mathematics

Mathematics

Problem

Let z1z_1 and z2z_2 be two distinct complex numbers and z=(1−t)z1+iz2z = (1-t)z_1 + iz_2 , for some real number tt with 0<t<10 < t < 1 and i=−1i = \sqrt{-1} . If arg⁡(w)\arg(w) denotes the principal argument of a non-zero complex number ww , then a. ∣z−z1∣+∣z−z2∣=∣z1−z2∣|z - z_1| + |z - z_2| = |z_1 - z_2| b. arg⁡(z−z1)=arg⁡(z−z2)\arg(z - z_1) = \arg(z - z_2) c. ∣z−z1zˉ−zˉ1z2−z1zˉ2−zˉ1∣=0\begin{vmatrix} z - z_1 & \bar{z} - \bar{z}_1 \\ z_2 - z_1 & \bar{z}_2 - \bar{z}_1 \end{vmatrix} = 0 d. arg⁡(z−z1)=arg⁡(z2−z1)\arg(z - z_1) = \arg(z_2 - z_1)

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