Question

If \(z = x + iy\), \(z^{1/3} = a - ib\) then \(\frac{x}{a} - \frac{y}{b} = k(a^2 - b^2)\), where \(k\) is equal to

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

\((a - ib)^3 = a^3 - 3a^2b i - 3ab^2 + ib^3\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Cube both sides and compare real and imaginary parts.
Step 2: Detailed Explanation:
\(z = (a - ib)^3 = a^3 - 3a^2 i b + 3a(ib)^2 - (ib)^3\)
= \((a^3 - 3ab^2) + i(-3a^2b + b^3)\)
So \(x = a^3 - 3ab^2 = a(a^2 - 3b^2)\), \(y = b^3 - 3a^2b = b(b^2 - 3a^2)\)
\(\frac{x}{a} - \frac{y}{b} = (a^2 - 3b^2) - (b^2 - 3a^2) = 4a^2 - 4b^2 = 4(a^2 - b^2) \rightarrow k = 4\)
Step 3: Final Answer:
\(k = 4\).