Complex Numbers And Quadratic Equations question 419
Question: If $ {{(x+iy)}^{1/3}}=a+ib, $ then $ \frac{x}{a}+\frac{y}{b} $ is equal to [IT 1982; Karnataka CET 2000]
Options:
A) $ 4(a^{2}+b^{2}) $
B) $ 4(a^{2}-b^{2}) $
C) $ 4(b^{2}-a^{2}) $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ {{(x+iy)}^{1/3}}=a+ib $
Þ $ (x+iy)={{(a+ib)}^{3}} $ $ =a^{3}+3a^{2}.ib+3a.{{(ib)}^{2}}+{{(ib)}^{3}} $ $ =a^{3}-3ab^{2}+i(3a^{2}b-b^{3}) $ Equating real and imaginary parts, we get $ \frac{x}{a}=a^{2}-3b^{2} $ and $ \frac{y}{b}=3a^{2}-b^{2} $
$ \therefore $ $ \frac{x}{a}+\frac{y}{b}=4(a^{2}-b^{2}) $
BETA
