Complex Numbers
Locus in complex plane
Grade 11
Question:
<p>Match the following. Consider the following for complex number \(z\):</p>
<table border='1'><tr><th>Column I</th><th>Column II</th></tr><tr><td>(A) \(\left||z - \cos^{-1}\cos 12| - |z - \sin^{-1}\sin 12|\right| = 8(\pi - 3)\)</td><td>(s) Locus of \(z\) is a portion of a line</td></tr><tr><td>(B) \(z^2 - i|z|^2 = k_2 - k_1\)</td><td>(r) Locus of \(z\) is a point of intersection of hyperbola</td></tr><tr><td>(C) \(\text{Re}(z+iy)^2 = \text{Re}(z+iy+x-iy)\)</td><td>(q) Rectangular hyperbola, eccentricity \(= \sqrt{2}\)</td></tr><tr><td>(D) \(\left|\dfrac{2z-i}{z+i}\right| = 2\)</td><td>(p) Straight line</td></tr></table>
<p>(A)→s, (B)→r, (C)→q, (D)→p</p>
<p>(A)→r, (B)→s, (C)→p, (D)→q</p>
<p>(A)→p, (B)→q, (C)→r, (D)→s</p>
<p>(A)→q, (B)→p, (C)→s, (D)→r</p>
Step-by-Step Solution
Key Concept: Evaluate inverse trig expressions using periodicity (cos⁻¹cos(12) = 12-3π, sin⁻¹sin(12) = 4π-12), then interpret the absolute value condition geometrically as a distance relationship in the complex plane.
<p><strong>Step 1: Evaluate inverse trig expressions</strong></p><p>Since 12 ≈ 3.82π and inverse functions have restricted ranges:</p><p>cos⁻¹(cos 12) = 2π(2) - 12 = 4π - 12 ≈ 0.434 (using periodicity in [0,2π])</p><p>sin⁻¹(sin 12) = (4π - 12) (using the identity)</p><p>Therefore: |12 - 3π| = 12 - 3π ≈ 2.575 and |4π - 12| ≈ 0.434</p><p><strong>Step 2: Simplify the locus equation</strong></p><p>||z - (12-3π)| - |z - (4π-12)|| = 8(π-3)</p><p>Let a = 12-3π and b = 4π-12, note that a + b ≈ π</p><p>The equation ||z-a| - |z-b|| = constant where the constant equals |a-b| defines a line (degenerate hyperbola)</p><p><strong>Step 3: Match (A) → (s)</strong></p><p>The locus is a <strong>portion of a line</strong> (the perpendicular bisector or line segment between two points)</p><p><strong>For complete matching:</strong></p><p>(B) z² - i|z|² = k₂ - k₁ → yields a hyperbola, likely (r)</p><p>(C) Re(z+iy)² = Re(z+iy+x-iy) → algebraic simplification gives rectangular hyperbola → (q)</p><p>(D) |2z-i|/|z+i| = 2 → apollonius circle/line transformation → (p) Straight line</p><p>∴ <strong>A→s, B→r, C→q, D→p</strong></p>
Correct Answer: A