4.4. The Fundamental Theorem of Algebra 139
as a simultaneous real systemax2 + bx - ay2 + c = (^0) w
y(2ax + b) = 0. CL)
(b) Show that the locus of (L) is a perpendicular pair of straight
lines intersecting in the point (-b/2a, 0).
(c) Verify that (H) can be rewritten in the form
(x + b/2a)2 - y2 = (1/2a)2(b2 - 4~).
(d) If b2 - 4ac = 0, show that the locus of (H) is a pair of per-
pendicular straight lines intersecting in (-b/2a,O). Sketch the
graphical solution of the two equations.
(e) Suppose b2 - 4ac # 0. Show that the locus of (H) is a hyperbola
whose center is (-h/2a,O) and whose asymptotes are the lines
x-y+b/2a=Oandz+y+b/2a=O.
(f) If b2 - 4ac > 0, show that the loci of (H) and (L) intersect in
two points on the x-axis. Sketch the graphical solution of the
two equations.
(g) If b2 - 4ac < 0, show that the loci of (H) and (L) intersect in
two distinct points on the line x = -b/2a which are symmetrical
about the real axis. Sketch the graphical solution of the two
equations.
(h) Show that (d), (f) and (g) confirm that:
(i) if b2 - 4ac = 0, the quadratic equation az2 + bz + c = 0 has
a single real root;
(ii) if b2 - 4ac > 0, the quadratic equation has two distinct real
roots;
(iii) if b2 - 4ac < 0, the quadratic equation has two distinct
nonreal roots, each the complex conjugate of the other.
(i) On the graphs sketched in (d), (f) and (g) draw a circle whose
center is at the origin and whose radius is sufficiently great that
its interior contains all the points of intersection of the loci of
(H) and (L). Label the points where the locus of (H) intersects
the circle by the letter R (<as in real) and the points where the
locus of(L) intersects the circle by the letter I (as in imaginary).
Verify that there are four points with each of the labels R and
I, and that the R-points alternate with the I-points.
- Illustrate graphically the solutions in the complex plane of the fol-
lowing equations: