41 Déterminer et représenter les ensembles de points M du plan dont l'affixe z vérifie la condition indiquée. a) |z+z̄-1|=4 b) |z-z̄-1+i|=2 c) arg(3i-z)≡0 [2π] d) arg(z-3+i)≡π/4 [π] e) arg(1/(z+2))≡π/2 [π] f) arg(z²-4)≡a

Step-by-step solution

1. 1

   Problem Statement

   We need to find the set of points M for each given condition.
2. 2

   Complex Number Basics

   Let $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part.

   $$z = x + iy$$
3. 3

   Substitute and Simplify

   Substitute $z = x + iy$ and $\bar{z} = x - iy$ into the equation and simplify.
4. 4

   Substitution

   Substitute $z$ and $\bar{z}$ into the equation.

   $$|(x+iy) + (x-iy) - 1| = 4$$
5. 5

   Simplify Expression

   Combine real and imaginary parts.

   $$|2x - 1| = 4$$
6. 6

   Solve for x

   The absolute value equation $|A|=B$ means $A=B$ or $A=-B$.

   $$2x - 1 = 4 \quad \text{or} \quad 2x - 1 = -4$$
7. 7

   Calculate x values

   Solve the two linear equations for $x$.

   $$2x = 5 \implies x = 5/2 \quad \text{or} \quad 2x = -3 \implies x = -3/2$$
8. 8

   Geometric Interpretation

   The set of points M are two vertical lines: $x = 5/2$ and $x = -3/2$.