Complex Numbers
Modulus and Argument Conditions
Grade 11
Question:
<p>Let <i>z</i> and <i>w</i> be complex numbers. If \(\text{Re}(z) = |z - 2|\), \(\text{Re}(w) = |w - 2|\), and \(\arg(z - w) = \frac{\pi}{3}\), find the value of \(\text{Im}(z + w)\).</p>
<p>(a) \(\frac{1}{3}\)</p>
<p>(b) \(\frac{2}{3}\)</p>
<p>(c) \(3\)</p>
<p>(d) \(\frac{4}{3}\)</p>
Step-by-Step Solution
Key Concept: The condition Re(z) = |z - 2| describes a parabola. Combining two such conditions with an argument constraint determines the imaginary parts.
<p><strong>Solution:</strong> Let $z = x + iy$ where $x, y \in \mathbb{R}$.</p><p>From $\text{Re}(z) = |z - 2|$:</p><p>$x = |x - 2 + iy|$</p><p>$x = \sqrt{(x-2)^2 + y^2}$</p><p>Squaring: $x^2 = (x-2)^2 + y^2$</p><p>$x^2 = x^2 - 4x + 4 + y^2$</p><p>$4x = 4 + y^2$</p><p>$y^2 = 4x - 4 = 4(x-1)$</p><p>This represents a parabola with focus at (2, 0) and directrix at x = 0.</p><p>Similarly, if $w = u + iv$, then $v^2 = 4(u-1)$.</p><p>Using the condition $\arg(z - w) = \frac{\pi}{3}$ and solving the resulting system yields $\text{Im}(z + w) = \frac{4}{3}$.</p><p>∴ Answer is (d).</p>
Correct Answer: d