Answers to Exercises; Chapter 1 2472.16. The abscissae 21 and 22 of the endpoints of the chord y = mx + k
are the roots of the equation (b2 + u2m2)x2 + 2u2kmx + u2(k2 - b2) = 0.
Hence the coordinates (X, Y) of the midpoints of the chord are given byX = -u2km(b2 + u2m2)-l, Y = mX + k = kb2(b2 + u2m2)-‘,from which it follows that the diameter lies along the line b2x + mu2y = 0.
For the case that all the chords are vertical, the locus has equation y = 0.
2.18. Since the quadratic equation has a nonrational root, p(u) # 0 for each
rational u. Since p(u) can be written as a fraction whose denominator is
q2, it follows that l/q2 5 Ip(u The right inequality follows from exercise
2(b). For the final assertion, take M = l/K.
3.1. Suppose x, y, u, v are real and x + yi = u + vi. Rearranging terms
and squaring yields (x - u)” = -(v - Y)~, whence x - u = v - y = 0.
3.2. (c) z - wz is a central similarity (dilatation) with factor ]w] followed
by a counterclockwise rotation through angle arg w.
3.4. Given z, to construct l/z: Construct the line A joining 0 to a point
u on the unit circle such that the real axis bisects the angle zOu. Let the
circle with center 0 and radius ]z] meet the positive real axis at T. Let B be
the line through I‘ and U, and C be the line through 1 parallel to B. The
intersection of A and C is the point l/z.
3.5. The locus evidently contains c/w = c-ii?/]w12. For z on the locus,
Re[(z - c/w)w] = 0, w h ence (z - c/w)w = ik for some real k. The locus is
a straight line consisting of points z = [c+ ik]Z?/lwl” (k E R).
3.6. Let z = x + yi. For k = 1, the locus is a straight line with equation
2x = - 1. For k # 1, the equation of the locus isx2 + y2 = k2(x2 + 2x + 1 + y2).For k > 1, the locus is a circle with center (-k2/(k2 - l),O) and radius
k/(k2 - 1). If k < 1, the locus is a circle with center (k2/(1 - k”),O) and
radius A/( 1 - k2).
3.7. Let the rocks U and V be the points 0 and 1 respectively in the complex
plane. If T is at z, then P is at iz and Q is at 1+ (-i)(z - 1) = (1+ i) - iz.