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Distance Formula Using Complex Numbers

If z = x + iy...

Question

If $z = x + i y, z^{1 / 3} = a - i b$ and $\frac{x}{a} - \frac{y}{b} = k (a^{2} - b^{2}),$ then $k =$

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Solution

$(x + i y)^{1 / 3} = a - i b$
$\Rightarrow x + i y = (a - i b)^{3} \Rightarrow x + i y = (a^{3} - 3 a b^{2}) + i (b^{3} - 3 a^{2} b)$
$\Rightarrow x = a^{3} - 3 a b^{2}, y = b^{3} - 3 a^{2} b$

$∴ \frac{x}{a} - \frac{y}{b} = a^{2} - 3 b^{2} - b^{2} + 3 a^{2} = 4 (a^{2} - b^{2})$

Hence $k = 4$

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Distance Formula Using Complex Numbers

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