Bonjour,
1)a)
z'A = 1/(-1/2+1) = 1/(1/2) = 2
z'B = 1/(-1/2 + i + 1) = 1/(1/2 + i) = (1/2 - i)/(1/4 + 1) = 4/5(1/2 - i) = 2/5 - 4i/5
z'C = .... = 2/(1 - i) = 1 + i
c) (z'B - z'A)/(z'C - z'A) = ..... = -2/5(-1-3i)
==> Arg[(z'B - z'A)/(z'C - z'A)] différent de 0
==> A', B', C' non alignés
2)
z = a + ib
|z| = V(a^2 + b^2)
|z + 1| = V((a+1)^2 + b^2)
|z| = |z+1|
==> a^2 + b^2 = (a+1)^2 + b^2
==> a^2 - (a+1)^2 = 0
==> (a - (a+1))(a + (a+1)) = 0
==> 2a + 1 = 0
==> a = -1/2
==> M appartient à (D)
Réciproquement : z = -1/2 + ib
==> |z| = V(1/4 + b^2) et |z+1| = V(1/4 + b^2)
b) |z'-1| = |1/(z+1) - 1| = |(1-z-1)/(z+1)| = |-z|/|z+1| = |z|/|z+1| = 1
c) |z'-1| = 1 ==> z' appartient à un cercle de centre I(1;0) et de rayon 1
