Complex Numbers
Lines in complex plane
Grade 11
Question:
<p><b>For Problems 20–22:</b> Consider the equation of line \(a\bar{z} + \bar{a}z + b = 0\), where \(b\) is a real parameter and \(a\) is fixed non-zero complex number.</p><p>The intercept of line on real axis is given by</p>
<p>(1) \(\dfrac{-2b}{a + \bar{a}}\)</p>
<p>(2) \(\dfrac{-b}{2(a + \bar{a})}\)</p>
<p>(3) \(\dfrac{-b}{a + \bar{a}}\)</p>
<p>(4) \(\dfrac{b}{a + \bar{a}}\)</p>
Step-by-Step Solution
Key Concept: The real axis intercept occurs when the imaginary part of z is zero (z = x, where x ∈ ℝ). Substitute z = x into the line equation and solve for x to find where the line crosses the real axis.
<p><strong>Step 1:</strong> The real axis intercept occurs when z is purely real, so let z = x where x ∈ ℝ.</p><p><strong>Step 2:</strong> Substitute z = x into the equation aāx + āax + b = 0</p><p>Since x is real and ā·a is real (product of complex conjugates), this gives:</p><p>(a·ā)x + (ā·a)x + b = 0</p><p><strong>Step 3:</strong> Simplifying: 2|a|²x + b = 0</p><p><strong>Step 4:</strong> Solving for x: x = -b/(2|a|²)</p><p>∴ The intercept on the real axis is <strong>-b/(2|a|²)</strong></p>
Correct Answer: A