If x-iy3-u-iv- then ux-vy is equal to

Given: (x iy)^3=u+iv ⇒ x^3 y^3(i)^3 3x^2yi+3xy^2(i)^2=u+iv ⇒ x^3+y^3i 3x^2yi 3xy^2=u+iv ⇒(x^3 3xy^2)+i(y^3 3x^2y)=u+iv On equating real and imaginary parts, w ...

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If x-iy3=u+iv...

Question

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

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

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

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

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

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Solution

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

$\Rightarrow x^{3} - y^{3} (i)^{3} - 3 x^{2} y i + 3 x y^{2} (i)^{2} = u + i v$
$\Rightarrow x^{3} + y^{3} i - 3 x^{2} y i - 3 x y^{2} = u + i v$
$\Rightarrow (x^{3} - 3 x y^{2}) + i (y^{3} - 3 x^{2} y) = u + i v$

On equating real and imaginary parts, we get

$u = x^{3} - 3 x y^{2} and v = y^{3} - 3 x^{2} y$
$\frac{u}{x} + \frac{v}{y} = \frac{x^{3} - 3 x y^{2}}{x} + \frac{y^{3} - 3 x^{2} y}{y}$
$= x^{2} - 3 y^{2} - 3 x^{2} + y^{2}$
$= - 2 x^{2} - 2 y^{2}$
$= - 2 (x^{2} + y^{2})$

$∴ \frac{u}{x} + \frac{v}{y} = - 2 (x^{2} + y^{2})$

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If (x−iy)3=u+iv, then ux+vy is equal to

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