First slide

Algebra of complex numbers

Question

If $z = x + i y and x^{2} + y^{2} = 16$ then the range of $| | x | - | y | |$ is

Moderate

A

[0, 4]
B

[0, 2]
C

[2,4]
D

none of these

Solution

Here $x = 4 \cos θ, y = 4 \sin θ$
$\begin{array}{l} ∴ ∥ x | - | y ∣ ∥ = | 4 | \cos θ | - 4 | \sin θ | | \\ = 4 | | \cos θ | - | \sin θ | | \\ = 4 \sqrt{1 - 2 | \cos 2 θ | | \sin θ |} \\ = 4 \sqrt{1 - | \sin 2 θ |} \end{array}$
Hence, the range is [0, 4].

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