Complex Numbers And Quadratic Equations question 635
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
Options:
A) 1
B) 2
C) 3
D) 4
Show Answer
Answer:
Correct Answer: D
Solution:
$ {z^{1/3}}=a-ib\Rightarrow z={{(a-ib)}^{3}} $
$ \therefore x+iy=a^{3}+ib^{3}-3ia^{2}b-3ab^{2}. $ Then $ x=a^{3}-3ab^{2}\Rightarrow \frac{x}{a}=a^{2}-3b^{2} $ $ y=b^{3}-3a^{2}b\Rightarrow \frac{y}{b}=b^{2}-3a^{2} $ So, $ \frac{x}{a}-\frac{y}{b}=4(a^{2}-b^{2}) $
BETA
