Question

Q. Find locus of points in plane satisfying given conditions.
(i) │z-1│=6
(ii) │z+3│+│z+1│= 4
(iii) Arg z=π/3
(iv) Arg(z-1) =-3 π/4

Answer 1

Answer on Question #57135 – Math – Complex Analysis

Find locus of points in plane satisfying given conditions.

(i) $|z - 1| = 6$

(ii) $|z + 3| + |z + 1| = 4$

(iii) $\operatorname{Arg} z = \pi / 3$

(iv) $\operatorname{Arg}(z - 1) = -3\pi / 4$

Solution

(i) $|z - 1| = 6$ represents a circle of radius 6 centered at point (1,0).

(ii) $|z + 3| + |z + 1| = 4$ represents an ellipse with foci at $(-3,0)$ and $(-1,0)$

(iii) $\operatorname{Arg} z = \pi / 3$ represents a straight line of the form

$$y = k x,$$

where $k = \tan \pi /3 = \sqrt{3}$ $x = \operatorname {Re}(z)$ $y = \operatorname {Im}(z)$

$$y = \sqrt {3} x$$

(iv) $\operatorname{Arg}(z - 1) = -3\pi /4$ represents a line of the form

$$y = k (x - 1),$$

where $k = \tan \left(-\frac{3\pi}{4}\right) = -1 / \sqrt{2}, x = \operatorname{Re}(z), y = \operatorname{Im}(z)$ .

$$y = - \frac {x - 1}{\sqrt {2}}$$

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