Complex Numbers
Locus in Complex Plane
Grade 11
Question:
<p>Locus of complex number satisfying \(\arg\left[\dfrac{(z - 5 + 4i)}{(z + 3 - 2i)}\right] = -\pi/4\) is the arc of a circle</p>
<p>(1) whose radius is \(5\sqrt{2}\)</p>
<p>(2) whose radius is 5</p>
<p>(3) whose length (of arc) is \(\dfrac{15\pi}{\sqrt{2}}\)</p>
<p>(4) whose centre is \(-2 - 5i\)</p>
Step-by-Step Solution
Key Concept: The argument condition arg[(z-z₁)/(z-z₂)] = θ represents a locus where the angle subtended by two fixed points z₁ and z₂ at any point z on the locus is constant, which traces an arc of a circle passing through both points.
<p><strong>Step 1:</strong> Let z = x + iy. The argument condition states:</p><p>arg[(z - 5 + 4i)/(z + 3 - 2i)] = -π/4</p><p><strong>Step 2:</strong> This can be rewritten as: arg[(z - (5 - 4i))/(z - (-3 + 2i))] = -π/4</p><p>This represents the locus of points z where the angle ∠AZB = -π/4 (directed angle), where A = 5 - 4i and B = -3 + 2i are fixed points.</p><p><strong>Step 3:</strong> By the argument property, the locus of points where arg[(z - z₁)/(z - z₂)] = constant is always an arc of a circle passing through z₁ and z₂.</p><p><strong>Step 4:</strong> The argument being negative (-π/4) indicates the direction of measurement, but the locus is still a circular arc.</p><p>To find the circle explicitly: Let (z - A)/(z - B) = re^{-iπ/4} where r > 0. This gives us a circle whose diameter endpoints and radius can be determined from the chord AB and the angle condition.</p><p>∴ The locus is <strong>an arc of a circle</strong>, confirming the statement is <strong>TRUE (Answer: 1)</strong></p>
Correct Answer: 1