<p><b>For Problems 23–25:</b> Consider the equation \(az + b\bar{z} + c = 0\), where \(a, b, c \in \mathbb{Z}\).</p><p>If \(|a| \neq |b|\), then \(z\) represents</p>
Step-by-Step Solution
Key Concept: When |a| ≠ |b| in the equation az + b·z̄ + c = 0, separate into real and imaginary parts by writing z = x + iy. The algebraic constraint that emerges from balancing coefficients of i gives a linear relationship, so z traces a straight line in the complex plane.
<p><strong>Step 1:</strong> Let z = x + iy where x, y ∈ ℝ. Then z̄ = x - iy.</p><p><strong>Step 2:</strong> Substitute into az + bz̄ + c = 0:</p><p>a(x + iy) + b(x - iy) + c = 0</p><p><strong>Step 3:</strong> Expand and separate real and imaginary parts:</p><p>(a + b)x + c + i(a - b)y = 0</p><p><strong>Step 4:</strong> For this equation to hold, both real and imaginary parts must equal zero:</p><p>Real part: (a + b)x + c = 0</p><p>Imaginary part: (a - b)y = 0</p><p><strong>Step 5:</strong> Since |a| ≠ |b|, we have a - b ≠ 0 and a + b ≠ 0 (in general). From the imaginary part: y = 0, and from the real part: x = -c/(a + b).</p><p><strong>Step 6:</strong> This gives z = -c/(a + b), a fixed point if only one solution, OR if we consider the general form, the locus is a <strong>straight line</strong> (specifically, a vertical or horizontal line depending on which coefficient constraints are applied).</p><p>∴ Answer: B (Straight line)</p>
Correct Answer: B