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Equality of 2 Complex Numbers

If u+ iv= x+ ...

Question

$If u + i v = (x + i y)^{3} then, \frac{u}{x} + \frac{v}{y} =$

A

4 (x^{2} - y^{2})

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B

4 (x^{2} + y^{2})

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C

2 (x^{2} - y^{2})

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D

2 (x^{2} + y^{2})

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Solution

The correct option is A $4 (x^{2} - y^{2})$
$u + i v = (x + i y)^{3}$
$= x^{3} + 3 x^{2} i y + 3 x i^{2} y^{2} + i^{3} y^{3}$
$= (x^{3} - 3 x y^{2}) + i (3 x^{2} y - y^{3})$

Equating real and imaginary parts, we get
$u = x^{3} - 3 x y^{2},$ $v = 3 x^{2} y - y^{3}$
$\Rightarrow \frac{u}{x} = x^{2} - 3 y^{2}, \frac{v}{y} = 3 x^{2} - y^{2}$
$∴ \frac{u}{x} + \frac{v}{y} = 4 (x^{2} - y^{2})$

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Equality of 2 Complex Numbers

Standard XII Mathematics

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