26 Chapter^1and then prove that lzl - Re z :5^1 / 2 iff z = ac with le - al :5 1. [S. I.
Drobnies, Amer. Math. Monthly, 71 (1970), 194).19. Determine the set all points z such that the ratio (1 - z)/(1 + z) is a
purely imaginary number.20. Show that the ratio z/(z - a)(l - az), where lal f; 1, represents a
positive real number whenever lzl = 1.21. If a and bf; 0 are complex numbers such that la+ bl = la - bl, prove
that ia/b is real.- If z1 + z2 + za = 0, show that:
(a) lz1 - z21^2 + lz2 - zal^2 + lza - z11^2 = 3(lz11
2
+ hl^2 + lzal
2
)and if z1 + z2 + za + Z4 = 0, then(b) lz1 - z21^2 + lz2 - zal^2 + lza - z41^2 + lz4 - z11^2 ;::: 2(lz1l^2 + lz21^2 +
lzal^2 + lz41^2 )where equality holds iff za = -z1 and Z4 = -z2.Remark (a) and (b) are special cases of Schoenberg's inequality: If
L~=l Zk = 0, then.n n
L lzk+i - zkl^2 ;::: 4 (sin^2 ; ) L lzkl^2
k=l k=l
where Zn+l = z1. [See Monthly, 57 (1950), 390-404.]- Let z1, z2, and w be complex numbers. Then w satisfies
iff w = c1z1 + c2z2 with lei I = lc2 I = 1. [H. A. MacLean, Amer. Math.
Monthly, 85 (1978), 105]
*24. (a) Prove that the equation az + bz + c = 0 represents a single point
in the complex plane if lal f; lbl. Find that point.
(b) Show that the equation above represents a straight line in the .com-
plex plane if lal = lbl f; 0 and ac =be. Show that in this case the
equation can always be put in the form Az + Az + C = 0, where
A f; 0 is complex and C is real.
(c) Show that the equation has no solution if lal = lbl and ac f; be
*25. Consider the equation zz + az + bz + c = 0.
(a) Show that the equation represents two points (distinct or not) inthe complex plane if a f; b.
(b) Show that the equation represents a circle if a = b, c is real and
lbl2 - c > 0.