Final answer:
The set of points in the complex plane that satisfy each of the given conditions are described individually. The solutions involve circles, straight lines, ellipses, and triangles. Depending on what the condition specifies, the set of points can be concentrated around particular shapes or areas.
Step-by-step explanation:
In order to describe the set of points in the complex plane that satisfy each of the given conditions, we need to analyze each condition individually.
a) Im z = -2:
This condition means that the imaginary part of z is -2. So, the set of points that satisfy this condition is all the complex numbers where the imaginary part is -2.
b) |z - 1 + i| = 3:
This condition means that the distance between the point z and the point (1, -1) in the complex plane is equal to 3. So, the set of points that satisfy this condition is a circle with center (1, -1) and radius 3.
c) |2z - i| = 4:
This condition means that the distance between the point 2z and the point (0, -1) in the complex plane is equal to 4. So, the set of points that satisfy this condition is a circle with center (0, -1/2) and radius 2.
d) |z - 1| = |z + i|:
This condition means that the distance between the point z and the point (1, 0) in the complex plane is equal to the distance between the point z and the point (0, 1). So, the set of points that satisfy this condition is a straight line passing through the points (1, 0) and (0, 1).
e) |z| = Re z + 2:
This condition means that the distance between the origin (0, 0) and the point z is equal to the real part of z plus 2. So, the set of points that satisfy this condition is a straight line passing through the origin (0, 0) with a slope of 1 and a y-intercept of 2.
f) |z - 1| + |z + 1| = 7:
This condition means that the sum of the distances between the point z and the points (1, 0) and (-1, 0) in the complex plane is equal to 7. So, the set of points that satisfy this condition is an ellipse with foci (1, 0) and (-1, 0) and a major axis of length 7.
g) |z| = 3|z - 1|:
This condition means that the distance between the origin (0, 0) and the point z is three times the distance between the point z and the point (1, 0). So, the set of points that satisfy this condition is an equilateral triangle with vertices at (0, 0), (1, 0), and (3, 0).
h) Re z < 0:
This condition means that the real part of z is negative. So, the set of points that satisfy this condition is all the complex numbers where the real part is negative.