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Properties of Modulus

a, b, c, d ∈ ...

Question

$a, b, c, d \in R$ such that $a^{2} + b^{2} = 4$ and $c^{2} + d^{2} = 2$ and if $(a + i b)^{2} = (c + i d)^{2} (x + i y)$ then $x^{2} + y^{2} =$

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Solution

Given that $(a + i b)^{2} = (c + i d)^{2} (x + i y)$
Taking modulus both the sides
$| (a + i b)^{2} | = | (c + i d)^{2} (x + i y) |$
$[∵ | z_{1} z_{2} | = | z_{1} | | z_{2} |, | z^{n} | = | z |^{n}]$
$∴ | a + i b |^{2} = | c + i d |^{2} | x + i y |$
$\Rightarrow a^{2} + b^{2} = (c^{2} + d^{2}) \sqrt{x^{2} + y^{2}}$
Given that $a^{2} + b^{2} = 4, c^{2} + d^{2} = 2$
$\Rightarrow 4 = 2 \sqrt{x^{2} + y^{2}}$
$\Rightarrow x^{2} + y^{2} = 4$

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a, b, c, d ϵ R

a^{2} + b^{2} = 4

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Properties of Modulus

Standard XII Mathematics

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