Let z1 and z2 be two complex numbers such that |z1| = 5 and (z1/z2)+(z2/z1)=1.
Find |z1-z2|2.
Note: The answer is not 75.
Let z_1 and z_2 be complex numbers such that |z_1| = 5 and (z_1/z_2) + (z_2/z_1) = 1. We are asked to find |z_1 - z_2|^2.
Step 1.
Let z_1 = 5e^{i\theta_1}, z_2 = 5e^{i\theta_2} (since their magnitudes are both 5).
Then
(z_1/z_2) + (z_2/z_1) = e^{i(\theta_1-\theta_2)} + e^{-i(\theta_1-\theta_2)} = 2 cos(\theta_1-\theta_2).
Given 2 cos(\theta_1-\theta_2) = 1, so cos(\theta_1-\theta_2) = 1/2. Thus (\theta_1-\theta_2) = 60^\circ or 300^\circ.
Step 2.
|z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2|z_1||z_2| cos(\theta_1-\theta_2)
= 25 + 25 - 255*(1/2)
= 50 - 25
= 25
Answer: 25
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