Complex Numbers
Geometry of complex numbers
Grade 11
Question:
<p>If \(|z_1 - 1| = 1\), \(|z_0 - 1| = 1\) and \(\left|\frac{z_1 - 1}{z_0 - 1}\right| = 1\), then which of the following are true?</p>
<p>\(\angle QCP = \frac{\pi}{2}\)</p>
<p>\(z_1 - 1 = (z_0 - 1)i\)</p>
<p>\(z_1 - 1 = -(z_0 - 1)i\)</p>
<p>\(|z_1| = |z_0|\)</p>
Step-by-Step Solution
Key Concept: Since |z₁ - 1| = |z₀ - 1| = 1, both z₁ and z₀ lie on a circle of radius 1 centered at 1. The condition |((z₁-1)/(z₀-1))| = 1 means these points are equidistant from the center, making (z₁-1)/(z₀-1) a point on the unit circle, so z₁ - 1 = e^(iθ)(z₀ - 1) for some real θ.
<p><strong>Step 1:</strong> From |z₁ - 1| = 1 and |z₀ - 1| = 1, both z₁ and z₀ lie on the circle C: |z - 1| = 1 (center at 1, radius 1).</p><p><strong>Step 2:</strong> The condition |(z₁-1)/(z₀-1)| = 1 means |z₁-1|/|z₀-1| = 1, which gives 1/1 = 1. This is automatically satisfied and provides no additional constraint—it's a redundant condition.</p><p><strong>Step 3:</strong> Therefore, any z₁ and z₀ on the circle |z - 1| = 1 satisfy all three conditions. This means:</p><ul><li>A: z₁ lies on circle |z - 1| = 1 ✓ TRUE</li><li>B: z₀ lies on circle |z - 1| = 1 ✓ TRUE</li><li>C: Both conditions hold for infinitely many pairs (z₁, z₀) on this circle ✓ TRUE</li></ul><p><strong>∴ Answer: A, B, C</strong></p>
Correct Answer: A, B, C