124 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS
(h) Observe that if a E C and r E �, the set of points a + reit, where 0 ::; t ::; 21t',
describes a circle with center at a and radius r.
(i) Let a, bE C. Prove-with a picture, if possible-that the area of the triangle with
vertices at O,a and b is equal to the one-half of the absolute value of Im(ab).
4.2.9 Less Easy Practice Problems. The following are somewhat more challenging.
Draw careful pictures, and do not be tempted to resort to algebra (except for checking
your work).
(a) It is easy to "simplify"a-bi
a+bi a^2 +b^2
by multiplying numerator and denominator by a - bi. But one can also verify
this without any calculation. How?(b) Iz + wi ::; Izl + Iwl, with equality if and only if z and w have the same direction or
point in opposite directions, i.e. if the angle between them is 0 or 1t'.(c) Let z lie on the unit circle; i.e., Izl = 1. Show that
Il - zl = 2sin
(ar;z)
without computation.(d) Let P(x) be a polynomial with real coefficients. Show that if z is a zero of P(x),
then z is also a zero; i.e., the zeros of a polynomial with real coefficients come in
complex-conjugate pairs.(e) Without much calculation, determine the locus for z for each of
Re (
z -I -i
) = 0 and 1m (
z - 1 - i
) = O.
z+l+i ' z+l+i
( f) Without solving the equation, show that all nine roots of (z -I)^10 = zlO lie on
the line Re ( z) = 1.
(g) Show that if Izl = I then
(h) Let k be a real constant, and let a, b be fixed complex numbers. Describe the
locus of points z that satisfy
(
z - a
arg )
z - b
= k.4.2.10 Grids to circles and vice versa. The problems below will familiarize you with
the lovely interplay among geometry, algebra, and analytic geometry when you ponder
complex transformations. The transformation that we analyze below is an example of
a Mobius transformation. See [29] for more details.
(a) Prove the following simple geometry proposition (use similar triangles).