Complex Numbers
Geometric Properties
Grade 11
Question:
<p>Let \(z_1, z_2\) and \(z_3\) be three distinct complex numbers, satisfying \(|z_1| = |z_2| = |z_3| = 1\). Which of the following is/are true:</p><p>(a) If \(\arg\left(\frac{z_1}{z_2}\right) = \frac{\pi}{2}\) then \(\arg\left(\frac{z - z_1}{z - z_2}\right) > \frac{\pi}{4}\) where \(|z| > 1\)</p><p>(b) \(|z_1z_2 + z_2z_3 + z_3z_1| = |z_1 + z_2 + z_3|\)</p><p>(c) \(\text{Im}\left(\frac{(z_1 + z_2)(z_2 + z_3)(z_3 + z_1)}{z_1 \times z_2 \times z_3}\right) = 0\)</p><p>(d) If \(|z_1 - z_2| = \sqrt{2}|z_1 - z_3| = \sqrt{2}|z_2 - z_3|\), then \(\text{Re}\left(\frac{z_3 - z_1}{z_3 - z_2}\right) = 0\)</p>
<p>(a) True/False</p>
<p>(b) True/False</p>
<p>(c) True/False</p>
<p>(d) True/False</p>
Step-by-Step Solution
Key Concept: Use properties of complex numbers on unit circle: $\bar{z} = 1/z$ and geometric interpretations of arguments and moduli.
<p><strong>Analysis of each option:</strong></p><p>(a) False: The argument relationship depends on specific position of z, not always satisfied.</p><p>(b) False: This equality does not hold in general for complex numbers on unit circle.</p><p>(c) True: Since $|z_i| = 1$, we have $\bar{z_i} = 1/z_i$. The expression $\frac{(z_1 + z_2)(z_2 + z_3)(z_3 + z_1)}{z_1z_2z_3}$ simplifies to a real number after conjugation check.</p><p>(d) True: The given chord lengths form an isosceles triangle. The points form a specific configuration where $\frac{z_3 - z_1}{z_3 - z_2}$ is purely imaginary.</p>
Correct Answer: c,d