If z=x+i y, z^ 1 / 3 =a-i b, then (x)/(a)-(y)/(b)=k (a^ 2 -b^ 2 ) where k is equal to

Correct answer: D. 4 z^ 1 / 3 =a-i b z=(a-i b)^ 3 x+i y=a^ 3 +i b^ 3 -3 i a^ 2 b-3 a b^ 2 x=a^ 3 -3 a b^ 2 (x)/(a)=a^ 2 -3 b^ 2 and y=b^ 3 -3 a^ 2 b (y)/(b)=b^ 2 -3 a^ 2 So, (x)/(a)-(y)/(b)=4 (a^ 2 -b^ 2 )

BITSAT 2014

Exam: BITSAT
Year: 2014
Subject: Mathematics
Chapter: Complex Number

BITSAT 2014 · Mathematics · Complex Number

If z=x+i y, z^ 1 / 3 =a-i b, then (x)/(a)-(y)/(b)=k (a^ 2 -b^ 2 )...

z=x+i y, z^{1 / 3}=a-i b,

then

\frac{x}{a}-\frac{y}{b}=k

\left(a^{2}-b^{2}\right)

where

k is equal to

Options

Correct answer

D. 4

Step-by-step solution

z^{1 / 3}=a-i b \Rightarrow z=(a-i b)^{3}\therefore x+i y=a^{3}+i b^{3}-3 i a^{2} b-3 a b^{2}\Rightarrow x=a^{3}-3 a b^{2} \Rightarrow \frac{x}{a}=a^{2}-3 b^{2} and

y=b^{3}-3 a^{2} b \Rightarrow \frac{y}{b}=b^{2}-3 a^{2}

So,

\frac{x}{a}-\frac{y}{b}=4 \left(a^{2}-b^{2}\right)

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