Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When dealing with complex numbers, manipulating their polar form can simplify the calculations for cube roots and ratios.

Answer 1

The Correct Option is A

The equation \( z = x + iy \) and \( z^{1/3} = a - ib \) leads to a relationship where the ratios \( \frac{x}{a} = \frac{y}{b} \) are equal to 1. Hence, the value of \( k = 1 \).
Final Answer: \[ \boxed{1} \]