<p>Let \(z = x - iy\) and \(z^{1/3} = p + iq\). If \(\dfrac{x}{p} + \dfrac{y}{q} = k(p^2 + q^2)\), then \(k\) is:</p>
Step-by-Step Solution
Key Concept: If z = x - iy and z^(1/3) = p + iq, then (p + iq)³ = x - iy. Expanding and comparing real and imaginary parts gives x = p³ - 3pq² and y = 3p²q - q³. Substitute these into the given expression to find k.
<p><strong>Step 1:</strong> Given z = x - iy and z^(1/3) = p + iq, so (p + iq)³ = x - iy.</p><p><strong>Step 2:</strong> Expand: (p + iq)³ = p³ + 3p²(iq) + 3p(iq)² + (iq)³ = p³ + 3p²qi - 3pq² - q³i = (p³ - 3pq²) + i(3p²q - q³).</p><p><strong>Step 3:</strong> Compare with x - iy:<br/>Real part: x = p³ - 3pq²<br/>Imaginary part: -y = 3p²q - q³, so y = q³ - 3p²q</p><p><strong>Step 4:</strong> Calculate x/p + y/q:<br/>x/p = (p³ - 3pq²)/p = p² - 3q²<br/>y/q = (q³ - 3p²q)/q = q² - 3p²<br/>Sum: (p² - 3q²) + (q² - 3p²) = p² + q² - 3q² - 3p² = -(2p² + 2q²) = -2(p² + q²)</p><p><strong>Step 5:</strong> Given x/p + y/q = k(p² + q²), we have:<br/>-2(p² + q²) = k(p² + q²)<br/>Therefore: k = -2</p><p>∴ Answer: B (k = -2)</p>
Correct Answer: B