Complex Numbers
Locus in complex plane
Grade 11
Question:
<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 \(|a| = |b|\) and \(\bar{a}c \neq b\bar{c}\), then \(z\) has</p>
<p>(1) infinite solutions</p>
<p>(2) no solutions</p>
<p>(3) finite solutions</p>
<p>(4) cannot say anything</p>
Step-by-Step Solution
Key Concept: When |a| = |b| in the equation az + b̄z + c = 0, the coefficients have equal magnitude. Writing z = x + iy and separating real/imaginary parts reveals that the equation represents a straight line (not a circle), which has infinitely many solutions.
<p><strong>Step 1:</strong> Let z = x + iy. Substitute into az + b̄z + c = 0.</p><p><strong>Step 2:</strong> Assume a = α + iβ and b = γ + iδ with |a| = |b|, so α² + β² = γ² + δ².</p><p><strong>Step 3:</strong> az + b̄z = (α + iβ)(x + iy) + (γ - iδ)(x - iy)</p><p>= (αx - βy) + i(βx + αy) + (γx + δy) + i(-δx + γy)</p><p>= (α + γ)x + (-β + δ)y + i[(β - δ)x + (α + γ)y] + c</p><p><strong>Step 4:</strong> For this to equal 0, both real and imaginary parts must vanish:</p><p>Real: (α + γ)x + (-β + δ)y + c = 0</p><p>Imaginary: (β - δ)x + (α + γ)y = 0</p><p><strong>Step 5:</strong> The condition ā·c ≠ b·c̄ ensures these two linear equations are consistent (not contradictory), and the system has infinitely many solutions forming a line in the complex plane.</p><p>∴ Answer: B (infinitely many solutions forming a line)</p>
Correct Answer: B