First slide

Geometry of complex numbers

Question

If $|ω| = 2,$ then the set of points $x + i y = ω - \frac{1}{ω}$ lie on

Moderate

A

circle
B

ellipse
C

parabola
D

hyperbola

Solution

$| ω | = 2 \Rightarrow ω = 2 e^{i θ} = 2 (\cos θ + i \sin θ)$
$\begin{array}{l} \Rightarrow x + i y = ω - \frac{1}{ω} = 2 e^{i θ} - \frac{1}{2} e^{- i θ} \\ \Rightarrow x = \frac{3}{2} \cos θ, y = \frac{5}{2} \sin θ \Rightarrow \frac{x^{2}}{9 / 4} + \frac{y^{2}}{25 / 4} = 1 \end{array}$
which represents an ellipse.

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