Complex Numbers
Circles and Locus in Complex Plane
Grade 11
Question:
<p>Let <span class="math">z_1</span> and <span class="math">z_2</span> be two complex numbers satisfying <span class="math">|z_1| = 9</span> and <span class="math">|z_2 - 3 - 4i| = 4</span>. Then, the minimum value of <span class="math">|z_1 - z_2|</span> is</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) <span class="math">2\sqrt{2}</span></p>
<p>(d) 0</p>
Step-by-Step Solution
Key Concept: Recognize that the minimum distance between two circles occurs along the line joining their centres. When one circle contains the other, calculate accordingly.
<p><strong>Solution:</strong></p><p>Clearly <span class="math">|z_1| = 9</span> represents a circle having centre <span class="math">C_1(0, 0)</span> and radius <span class="math">r_1 = 9</span>.</p><p><span class="math">|z_2 - 3 - 4i| = 4</span> represents a circle having centre <span class="math">C_2(3, 4)</span> and radius <span class="math">r_2 = 4</span>.</p><p>The minimum value of <span class="math">|z_1 - z_2|</span> equals the minimum distance between the two circles.</p><p>Distance between centres: <span class="math">|C_1C_2| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5</span></p><p>Minimum distance = <span class="math">|C_1C_2| - r_1 - r_2 = 5 - 9 - 4 = -8</span></p><p>Since this is negative, one circle is inside the other. The minimum distance is <span class="math">|r_1 - |C_1C_2|| - r_2 = |9 - 5| - 4 = 4 - 4 = 0</span></p><p><strong>∴ The answer is (d) 0</strong></p>
Correct Answer: d