1.3 • THE GEOMETRY OF COMPLEX NUMBERS 21- Sketch the sets of points determined by the following relations.
(a) lz + 1 - 2il = 2.
(b} Re(z+ 1) = 0.
(c) lz + 2il $ 1.
(d} Im(z -2i} > 6.- Prove that v'2 lzl;:: IRe(z)I + IIm(z)I.
8. Show that the point •^1 t•^2 is the midpoint of the line segment joining z1 to z2.
- Show that lz1 - z2I $ lzil + lz2I.
- Prove that lzl = 0 iff z = O.
- Show that if z f. 0, the four points z, z, -z, and -:z are the vertices of a rectangle
with its center at the origin.
12 ~ Show t hat if z f. 0, the four points z, iz, -z, and -iz are the vertices of a square
with its center at the origin.
Show that the equation of the li ne through the points z 1 and z 2 can be expressed
in the form z = z 1 + t(z2 - z 1 }, where t is a real number.
Show that the nonzero vectors z 1 and z2 are parallel iff Im(z 1 z:i) = 0.
Show that lz1z2z3I = lzil lz2l lz3j.
Show that lz"I = lzl", where n is an integer.
Suppose t hat either lzl = I or lwl = 1. Prove that lz -w l = ll - zwl.
Prove the Cauchy- Schwarz inequality: IE ZkWk' $
k=I
Show llz1l - lz2ll $ lz1 -Z2I·
Show that z 1z2 + Z1Z2 is a real number.
If you study carefully the proof of the triangle inequality, you will not e that the
reasons for the inequality hinge on Re(z1z2) $ lz1z2I· Under what conditions will
these two quantities be equal, thus turning the triangle inequality into an equality?
Prove that lz1 -z2 12 = lz1 12 - 2 Re(z1z:i) + lz2 12.
Use induction to prove t hat lk~l Zkl $ k~
1
lzkl for all natural numbers n.- Let z1 and z2 be two d istinct points in t he complex plane, and let K be a positive
real constant that is less than the distance between z 1 and z2.
(a) Show that the set of points {z : lz - z1 I - lz - z2I = K} is a hyperbola
with foci z 1 and z2.
(b) Find the equation of the hyperbola with foci ± 2 that goes through the
point 2 + 3i.
( c) Find the equation of t he hyperbola with foci ±25 that goes through
the point 7 + 24i.