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Solutions to Problems; Chapter (^4 319)
squares of the lengths of AP, BP and CP, where P is any point distant
4, &, fi from BC, CA, AB respectively. Since there is exactly one such
point, the system has exactly one solution.
8.34. Suppose abc # 0. Then xyz # 0 and the system becomes y-l +z-l =
a-‘, etc., whence 2x-l = b-’ + c-l - a-‘, etc. If, say a = 0 and bc # 0,
then yz = 0 + x = 0 =+ y = z = 0. If, say a = b = 0 and c # 0, then
either z = 0 and x, y are arbitrary solutions of the third equation or else
x = y = 0 and t is arbitrary. If a = b = c = 0, then two of x, y, z must
vanish and the third is arbitrary.
9.1. For each complex z, z and -z - l/a have the same image under
the mapping. Hence the mapping is one-one on the closed unit disc iff
IzI 5 1 * 1% + l/al > 1. T a k ing z = 0 yields the necessary condition
Ial < 1. If further, (al # l/2, then taking z = -IuI/a # -1/2a yields the
condition Ial < l/2. Hence, it is necessary that Ial 2 l/2.
But this condition is also sufficient. The intersection of the closed discs
with radii 1 and centers 0 and -l/a is empty if Ial < l/2 and contains
a single point if Ial = l/2. Hence, for Ial 5 l/2, the conditions 1.~1 5 1,
1% + l/al 5 1 are simultaneously fulfilled iff 101 = l/2 and .z = --z - l/a =
-1/2a.
9.2. If the straight line passes through the origin, its image under t - z2
is a ray emanating from the origin and not a parabola. So we exclude this
case. Let w be the point on the line closest to the origin. Then a typical
point on the line is represented by w + iwr = w( 1 + ir), where r is real. The
image of this point is w”( 1 - r2 + 2ir). The locus of 1 - r2 + 2ir is a parabola
whose axis coincides with the real axis and whose vertex is at 1. The locus
of w”( 1 - r2 + 2ir) is the image of this parabola under a dilatation followed
by a rotation, and so is a parabola whose vertex is at w2.
9.3. Three points in C are vertices of an equilateral triangle iff they can be
obtained from the points 1, w, w2 by a dilatation followed by a translation.
The center of the triangle is represented by one third of the sum of the
numbers corresponding to its vertices.