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Addition of Vectors

If 𝑥+𝑖𝑦=√�...

Question

If $x + i y = \sqrt{\frac{a + i b}{c + i d}}$ , then write the value of $(x^{2} + y^{2})^{2}$ .

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Solution

$∵ x + i y = \sqrt{\frac{a + i b}{c + i d}}$

Taking modulus of both sides
$\Rightarrow | x + i y | = ∣ ∣ ∣ \sqrt{\frac{a + i b}{c + i d}} ∣ ∣ ∣$

$\Rightarrow | x + i y | = {∣ ∣ ∣ \frac{a + i b}{c + i d} ∣ ∣ ∣}^{\frac{1}{2}}$

Squaring both sides
$\Rightarrow∣ x + i y ∣^{2} = (\frac{∣ a + i b ∣}{∣ c + i d ∣})$

$\Rightarrow x^{2} + y^{2} = \frac{\sqrt{a^{2} + b^{2}}}{\sqrt{c^{2} + d^{2}}}$

Again, squaring both sides
$\Rightarrow (x^{2} + y^{2})^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}$

$∴ (x^{2} + y^{2})^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}$

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Addition of Vectors

Standard XII Mathematics

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