§9. Geometry and Trigonometry
9.1. (a) Sketch the parabola with equation y2 = 4x. Consider the
family of parallel chords with equation
y = mx + b, where m is a fixed parameter and k is allowed to vary. Argue that
the midpoint of the
chord of equation y = mx + b is given by (X, Y ) where
and Y = mX = b, with
and
the two solutions of the quadratic equation
or
(b) Without solving the quadratic equation in (a), use the
relationship between the coefficients and
roots to obtain an ex pression for X . Show that Y does not depend on b. What does
this tell you about
the locus of (X, Y )?
(c) Redo parts (a) and (b) by setting up an equation in y rather than x and
computing
directly.
9.2. A diameter of a conic section is the locus of the midpoints of a family of
parallel chors.
(a) Sketch the ellipse with equation (x2/9)+(y2/4) = 1
along with some chords in the family y = x+k
where k is a parameter. (This could be d one with a calculator or with some
geometric computer software.
In the latter case, try to trace the midpoints of the chords.)
(b) Follow the strategy used in Exercise 16 to show that the locus of the
midpoints of the chords is a
straight line.
(c) Generalize to the general conic section of equation
Corroborate your findings by taking particular choices of
coefficients and chord slopes and graphing
them with a computer or calculator.
9.3. Suppose that, in a triangle ABC, one angle B and two sidelengths a = |BC|
and b = |AC| are
known. What is the length of the remaining side? One way to obtain this is to
use the Law of Cosines
to obtain c = |AB|. Let us rewrite this
third sidelength as a variable x and
arrange the equation to
This is a quadratic equation, and so will have two
solutions, which could be real or nonreal, positive or
negative, or coincident. In this exercise, we will see how this relates to the
geometry of the situation.
(a) Verify that the discriminant D of the quadratic in (*) is 4(b2 −
a2 sin2 B). Explain why D is
non negative if and only if a, b and B correspond to data for a feasible
triangle. What happens if D = 0?
Explain how the geometry supports the fact that (*) has a single solution is
this case.
(b) Suppose that a, B and B are data for a feasible triangle. By considering the
sum of the roots,
explain why (*) has at least one positive solution.
(c) Determine conditions of a and b that (*) has (i) exactly one, (ii) exactly
two, positive solutions.
Relate this to the geometric possibilities for the triangle. In the case where
there is a negative solution,
explain how it might be interpreted.
9.4. Let a, b, c be real numbers. We consider solutions of the quadratic
equation az2 + bz + c = 0 where
z = x + yi is a complex number.
(a) Show that the complex equation az2 + bz + c
= 0 is equivalent to the system of real equations:
(b) Considering (2) in the form y(ax + b) = 0, describe
its locus.
(c) Show that (1) can be written in the form
Describe the locus of this equation in the three cases:
(i) b2 = 4ac; (ii) b2 > 4ac; (iii) b2 < 4ac.
(d) The solutions of the system (1) and (2) are represented in the plane by
points (x, y) that lie on the
intersection of the loci of (1) and (2). When b2 = 4ac, show that
there is a single such point and that it
lies on the real axis. When b2 > 4ac, show that there are two points
on the real axis, each a reflection
of the other in the line Re z = −b/2a. When b2 < 4ac, show that there
are two points not on the real
axis that are mirror images of each other with respect to the real axis. Explain
how this is consistent
with what you already now about real and imaginary roots of a quadratic.
§10. Approximation.
10.1. Let c be a positive real number. A standard way to approximate the square
root of c is to begin with a
positive guess u and then proceed to a new guess
This is repeated over and over until
the desired degree of approximation is reached.
(a) Verify that if c = 2 and the first guess is 1, then this process yields the
sequence of approximants:
(where the decimals forms are not exact).
(b) Use the process to approximate
(c) Show that
if and only if
and that
if and only if
. Noting that v is
the average of u and c/u, explain why it is reasonable to expect that v might be
a better approximation
that u.
(d) Verify that
Deduce that every approximation beyond the first exceeds
,
and prove that from this point on the
sequences decreases. Why does the sequence tend towards
?
10.2. We look at the geometry of the situation of Exercise 10.1. As before, we
have that c > 0.
(a) Let x > 0. Use the Arithmetic-Geometric Means Inequality (Exercise 4.1) to
prove that
with equality if and only if
(b) Verify that
Use this to argue that
is a decreasing function of x for
and an increasing function
of x for
(c) With the same axes, sketch the graphs of both of the curves y = x and
for x>0
Where do these curves intersect? What are the asymptotes of the second curve?
(d) Using the graphs in (c), we can illustrate the behaviour of the
approximating sequence for
described in Exercise 10.1. Let
be the first approximant. Locate on your
sketch a possible
position of
. Let
Locate
and
. These three points will
be on the respective curves
, y = x and y = 0. We continue on in this
way. Suppose that
un has been found. Let
Locate
and
Describe from your diagram what eventually
happens to the terms of the sequence
.
10.3. The recursion of Exercise 10.1 can be defined when c is negative, even
through c does not have a real
square root in this case. What will happen? To focus the discussion, consider
the case c = −1.
(a) Sketch the curve
for real non zero x , and attempt an anlysis as in Exercise
10.2.(d), using various starting points. In this
case, you may find it helpful to use a calculator or computer to generate the
terms of the sequence of
“approximants”, or even to use the computer to draw the whole situation for you.
(b) To get a handle on the situation, we note that any real number can be
written in the form x = cot θ
for some number θ lying strictly between 0 and π. Consider the transformation
If x = cot θ, show that the image of x under this
transformation is cot 2θ. Thus, in terms of the
mapping is conjugate (essentially the same in its mathematical structure) to U:
θ→ 2 (modulo π)
(this simply means that if you add, subtract two angles or multiply by a
constant, you add an integral
multiple of π to put the result of the ope ration in the interval (0, π) using a
kind of “clock arithmetic”).
(c) Does the transformation T have any fixed points? (These are points x for
which T(x) = x. You can
answer this question directly, but also by looking at the mapping U and
reinterpreting what you find in
terms of T.)
(d) Let
and for n ≥3, let
Determine a simple expression for
(e) Does the transformation T have any points of period 2? (This asks whether
there are any numbers
u for which T(u) = v for some number v and T(v) = u, so that two applications of
the mapping T take
the point back to itself.) Answer this question directly by looking at the
equation
T(T(x)) = x .
Now answer it by working through the operator U. For what values of θ does U(U(θ )
= 4θ differ from
θ
by a multiple of π. Are your results consistent?
(f) A point p is a point of period k for T if and only if
, where
and
for k ≥2. Either directly or working through
the operator U, determine if T has points of
period k for k is a positive integer exceeding 1. Use a calculator to work out
the approximate values of
such points and check the result by applying the operator T.
§11. The logistic dynamical system.
We suppose that k is a positive parameter and define the function
for 0 ≤x ≤1. We
can use
to define a dynamical system as follows :
Begin with any point
in the closed
interval
For each nonnegative integer n,
define
11.1. One can use graphical methods in helping us visualize how the sequence
defined for the dynamical
system behaves. Suppose that we have a sketch of the curves with equations
y = pk(x)
and
y = x .
For each nonnegative integer n, plot the points
and
. By drawing lines
parallel to the axes and making use of the line y = x, indicate geometrically
how the point
can be found. Thus, we can indicate on the x−axis the progress of the sequence
.
11.2. Consider the case 0 < k < 1. Sketch the curves as indicated in (a) and use
your diagram to argue that
Verify this analytically, by first verifying that
whenever
11.3. Suppose that k > 1. Determine a number u for which 0 < u < 1 and
11.4. Consider the case 1 < k < 2. Sketch the curves as in (a), being careful to
indicate on which side of the
line
the curves intersect. Analyze the
types of behaviour of the sequence for values of
in the
closed interval [0, 1].
11.5. Consider the case 2 < k < 3. Sketch the curves as in (a) and analyze the
behaviour or sequences
.
Verify that
and use this to check that, when
and
have opposite signs and
Analyze the behaviour of the sequence
for various cases of
in [0, 1].
11.6. Let k > 1 and let u be as defined in part (c). Determine
in terms of k, where
denotes the
derivative of
. Prove that
if and only if 1 < k < 2. What effect
do you think that the
value of the derivative of
at u has on the
behaviour of sequences
that start off with a value
close to u?
11.7. We study the possibility of sequences
of period 2, i.e., there are two distinct values v and w for
which
when n is even and
when n is odd, so that the sequence proceeds
{u, v, u, v, · · ·}.
To do this, we define the second iterate of
:
Determine the polynomial
and specify its degree. Prove that if
and
then
and
11.8. To solve the equation
, we can write it in the form
Explain why
is a factor of the left side, and use this
fact to write the left side as a product of
quadratics. Thus determine v and w.
11.9. For the cases 1 < k < 2, 2 < k < 3, k = 3 and k < 3, show on a graph the
location of v and w.
11.10. Investigate the behaviour of the sequence
when k > 3. You may
find a pocket calculator of some
use in this enterprise.
§12. Miscellaneous
12.1. Let ABCD be a cyclic quadrilateral with side AD of
length d, with d the diameter of the corcum circle
of ABCD. Suppose that AB and BC both have length a while CD has length b. We are
given that a,
b and d are three positive integers.
(a) Prove that d cannot be a prime number, nor twice an odd prime number.
(b) What is the minimum integral value of d that admits the given configuration?
