Loci in the complex plane

Question 1

z - z1 | = r

Answer 1

circle centred z1 with radius r

Question 2

arg(z - z1) = theta

Answer 2

half-line starting at z1 with argument theta

Question 3

z - z1 | = | z - z2|

Answer 3

perpendicular bisector of line between z1 and z2

Question 4

z - z1 | = k | z - z2 |, where k ≠ 1

Answer 4

circle with a diameter that lies on the line between z1 and z2 (one end of the diameter lies between z1 and z2 and the other lies outside)

Question 5

arg((z - z1)/(z - z2)) = theta

Answer 5

arc of a circle through z1 and z2. The arc is always drawn counterclockwise from z1 to z2. If theta < pi/2 the major arc is drawn, otherwise minor arc.

Question 6

z - z1 | + | z - z2 | = c

Answer 6

Ellipse with major axis along the line between z1 and z2 (extend the line and find the two symmetric points that satify the locus).

Minor axis lies along the perpendicular bisector of the major axis, so find the two points on the bisetor which satisfy the locus.