First slide

Geometry of complex numbers

Question

If $z \neq 0$ lies on the circle $| z - 1 | = 1$ and $ω$ $= 5 / z,$ then $ω$ lies on

Moderate

A

a circle
B

an ellipse
C

a straight line
D

a parabola

Solution

$z = 5 / ω$ and $| z - 1 | = 1$
$\Rightarrow |\frac{5}{ω} - 1| = 1 \Rightarrow | ω - 5 | = | ω |$
that is, $ω$ lies on the perpendicular bisector of the segment joining 0 and $5 + 0 i$
Therefore $ω$ lies on the straight line $Re (ω) = 5 / 2$

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