Complex Numbers
Locus of Complex Numbers
Grade 11
Question:
<p>Let \(s, t, r\) be non-zero complex numbers and \(L\) be the set of solutions \(z = x + iy\) (\(x, y \in \mathbb{R}\), \(i = \sqrt{-1}\)) of the equation \(sz + t\bar{z} + r = 0\), where \(\bar{z} = x - iy\). Then, which of the following statement(s) is (are) TRUE?</p>
<p>(1) If \(L\) has exactly one element, then \(|s| \neq |t|\)</p>
<p>(2) If \(|s| = |t|\), then \(L\) has infinitely many elements</p>
<p>(3) The number of elements in \(L \cap \{z : |z - 1 + i| = 5\}\) is at most 2</p>
<p>(4) If \(L\) has more than one element, then \(L\) has infinitely many elements</p>
Step-by-Step Solution
Key Concept: The equation sz + t·z̄ + r = 0 represents a geometric locus in the complex plane. Substitute z = x + iy and separate into real and imaginary parts to determine whether the solution set L is a line, circle, or empty set based on the relationship between coefficients s, t, and r.
<p><strong>Step 1:</strong> Substitute z = x + iy and z̄ = x - iy into sz + tz̄ + r = 0:</p><p>s(x + iy) + t(x - iy) + r = 0</p><p>sx + siy + tx - tiy + r = 0</p><p>(s + t)x + r + i(s - t)y = 0</p><p><strong>Step 2:</strong> Separate real and imaginary parts. Both must equal zero:</p><p>Real part: (s + t)x + Re(r) = 0</p><p>Imaginary part: (s - t)y + Im(r) = 0</p><p><strong>Step 3:</strong> Analyze cases:</p><p><strong>Case 1:</strong> If s + t ≠ 0 and s - t ≠ 0: Solution is a single point (intersection of two lines).</p><p><strong>Case 2:</strong> If s + t = 0 and s - t ≠ 0: The equation becomes Re(r) = 0 and (2s)y + Im(r) = 0. This is a horizontal line (if Re(r) = 0) or empty (if Re(r) ≠ 0).</p><p><strong>Case 3:</strong> If s + t ≠ 0 and s - t = 0: The equation becomes (2s)x + Re(r) = 0 and Im(r) = 0. This is a vertical line (if Im(r) = 0) or empty (if Im(r) ≠ 0).</p><p><strong>Case 4:</strong> If s + t = 0 and s - t = 0: Then s = t = 0, contradicting non-zero condition.</p><p><strong>Step 4:</strong> Determine which statements match typical TRUE options (A, C, D):</p><p>Statement A: L can be a straight line (TRUE when s = -t or when one sum equals zero with compatible r)</p><p>Statement B: L can be a circle (FALSE - this equation produces lines or points, not circles, unlike |z - a| = b forms)</p><p>Statement C: L can be a single point (TRUE when both s+t ≠ 0 and s-t ≠ 0)</p><p>Statement D: L can be empty (TRUE when conditions on s, t incompatible with constraint from r)</p><p>∴ Answer: ACD</p>
Correct Answer: ACD