First slide

Geometry of complex numbers

Question

$A (z_{1}), B (z_{2}), C (z_{3}) & D (z_{4}) are four points in complex plane where z_{1}, z_{2}, z_{3} & z_{4} are roots of the equation z^{4} + z^{3} + 2 z^{2} + z + 1 = 0 then$

Difficult

A

ABCD is rectangle
B

ABCD is cyclic
C

$|z_{1}| = |z_{2}| = |z_{3}| = |z_{4}| = 1$
D

$(z_{1}^{4} + z_{2}^{4} + z_{3}^{4} + z_{4}^{4})$ is purely imaginary

Solution

${(z^{2} + 1)}^{2} + z (z^{2} + 1) = 0 \Rightarrow (z^{2} + 1) (z^{2} + z + 1) = 0$
$roots are i, - i, ω & ω^{2}$

magnitudes are equal for all the above complex numbers and each is equal to 1, and they are concyclic

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