First slide

Geometry of complex numbers

Question

$If | 2 z - 4 - 2 i | = | z | \sin (\frac{π}{4} - \arg z), then locus of z is/an$

Difficult

A

ellipse
B

circle
C

parabola
D

pair of straight line

Solution

$\begin{array}{l} Let z = x + i y = r (\cos θ + i \sin θ), \\ Then the equation is | 2 (x - 2) + 2 i (y - 1) | = r (\frac{1}{\sqrt{2}} \cos θ - \frac{1}{\sqrt{2}} \sin θ) \\ = \frac{1}{\sqrt{2}} (r \cos θ - r \sin θ) \\ \Rightarrow \sqrt{(x - 2)^{2} + (y - 1)^{2}} = \frac{1}{2} (\frac{x - y}{\sqrt{2}}) \\ It is an ellipse with focus at (2, 1) and directrix x - y = 0 \end{array}$
$and eccentricity = \frac{1}{2} < 1$
The locus represents an ellipse
Therefore, the correct answer is (1).

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