Complex Numbers And Quadratic Equations question 450
Question: If $ z=x+iy,{z^{1/3}}=a-ib $ and $ \frac{x}{a}-\frac{y}{b}=k(a^{2}-b^{2}) $ then value of k equals [DCE 2005]
Options:
A) 2
B) 4
C) 6
D) 1
Show Answer
Answer:
Correct Answer: B
Solution:
$ {{(x+iy)}^{1/3}}=a-ib $ $ x+iy={{(a-ib)}^{3}}=(a^{3}-3ab^{2})+i(b^{3}-3a^{2}b) $
Þ $ x=a^{3}-3ab^{2},y=b^{3}-3a^{2}b $
Þ $ \frac{x}{a}=a^{2}-3b^{2},\frac{y}{b}=b^{2}-3a^{2} $
$ \therefore $ $ \frac{x}{a}-\frac{y}{b}=a^{2}-3b^{2}-b^{2}+3a^{2} $ $ \frac{x}{a}-\frac{y}{b}=4(a^{2}-b^{2})=k(a^{2}-b^{2}) $
$ \therefore $ $ k=4 $ .
BETA
