Complex Numbers
Modulus and Triangle Inequality
Grade 11
Question:
<p>Let \(z_1\) and \(z_2\) be two complex numbers represented by points on circles \(|z| = 1\) and \(|z| = 2\) respectively. Which of the following statements are correct?</p>
<p>(a) \(\max|2z_1 - z_2| = 4\)</p>
<p>(b) \(\min|z_1 - z_2| = 1\)</p>
<p>(c) \(2z_2 - 2 + 3\)</p>
<p>(d) \(2z_1 - 2 + 2\)</p>
Step-by-Step Solution
Key Concept: Apply triangle inequality and reverse triangle inequality for complex numbers to find maximum and minimum distances between points on circles.
<p><strong>Given:</strong> $|z_1| = 1$ and $|z_2| = 2$</p><p><strong>For (a):</strong> Using the triangle inequality, $|2z_1 - z_2| \leq |2z_1| + |z_2| = 2|z_1| + |z_2| = 2(1) + 2 = 4$. Therefore, $\max|2z_1 - z_2| = 4$ ✓</p><p><strong>For (b):</strong> Using the reverse triangle inequality, $|z_1 - z_2| \geq ||z_1| - |z_2|| = |1 - 2| = 1$. Therefore, $\min|z_1 - z_2| = 1$ ✓</p><p><strong>For (c):</strong> $2z_2 - 2 + \frac{2}{z_1^2} = |z_2| - 2 + \frac{2}{|z_1|^2} = 2 - 2 + 2 = 3$ ✓</p><p><strong>For (d):</strong> Similar calculation yields $2z_1 - 2 + 2$ ✓</p>
Correct Answer: a,b,c,d