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INTRODUCTION TO
ANALYTIC GEOMETRY
BY
PEECEY R SMITH,
PH.D.
N
PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL
YALE UNIVERSITY
AND
AKTHUB SULLIVAN
GALE, PH.D.
ASSISTANT PROFESSOR OF MATHEMATICS IN
THE UNIVERSITY OF ROCHESTER
GINN & COMPANY
BOSTON
NEW YORK CHICAGO LONDON
COPYRIGHT,
1904, 1905,
BY
ARTHUR SULLIVAN GALE
ALL BIGHTS RESERVED
65.8
GINN & COMPANY
PRIETORS
BOSTON
PROU.S.A.
PEE FACE
In preparing this volume the authors have endeavored to write
a drill book for beginners which presents, in a manner conforming with modern ideas, the fundamental concepts of the subject.
The subject-matter
is
more than the minimum required
much more as is necessary to permit
slightly
for the calculus, but only as
of
some choice on the part of the teacher.
text
is
It is believed that the
complete for students finishing their study of mathematics
with a course in Analytic Geometry.
The authors have
form of a
treatise
intentionally avoided giving the book the
on conic
sections.
Conic sections naturally
appear, but chiefly as illustrative of general analytic methods.
Attention
is
called to the
method of treatment.
The
subject
is
developed after the Euclidean method of definition and theorem,
without, however, adhering to formal presentation.
tage
is
obvious, for the student is
of each acquisition.
examples are
This
is
worked out
The advan-
sure of the exact nature
Again, each method
stated in consecutive steps.
illustrative
made
is
summarized in a rule
a gain in clearness.
Emphasis has everywhere been put upon the analytic
is, the student is taught to start from the equation.
that
shown how
to
to use it in
any other way.
work with the
Many
in the text.
figure as a guide, but is
Chapter III
may
side,
He
is
warned not
be referred to in
this connection.
The
etry
is
object of the
two short chapters on Solid Analytic Geom-
merely to acquaint the student with coordinates in space
iii
20206fi
PREFACE
iv
and with the
relations
between surfaces, curves, and equations in
three variables.
Acknowledgments are due to Dr. W. A. Granville for many
helpful suggestions, and to Professor E. H. Lockwood for suggestions regarding
NEW
some of the drawings.
HAVEN, CONNECTICUT
January, 1905
CONTENTS
CHAPTER
I
REVIEW OF ALGEBRA AND TRIGONOMETRY
PAGE
SECTION
1.
Numbers
2.
Constants
...........
.......
3.
The
4.
Special quadratics
5.
Cases
6.
Variables
7.
Equations in several variables
Functions of an angle in a right triangle
8.
quadratic.
when
Typical form
.
.
5
.
9
9
.
.
11.
12.
Rules for signs
13.
Greek alphabet
10.
1
2
4
the roots of a quadratic are not independent
Angles in general
Formulas and theorems from Trigonometry
Natural values of trigonometric functions
9.
1
..."
.....
.
.
.
.
11
11
....
.
.
.
.
12
.
14
15
15
CHAPTER
II
CARTESIAN COORDINATES
14.
Directed line
16
15.
Cartesian coordinates
17
16.
Rectangular coordinates
18
17.
Angles
21
18.
Orthogonal projection
22
19.
Lengths
20.
Inclination
24
and slope
27
21.
Point of division
31
22.
Areas
35
23.
Second theorem of projection
.
40
CONTENTS
vi
CHAPTER
III
THE CURVE AND THE EQUATION
SECTION
PAGE
24.
Locus of a point satisfying a given condition
25.
26.
Equation of the locus of a point satisfying a given condition
First fundamental problem
27.
General equations of the straight line and
28.
Locus of an equation
^
Second fundamental problem
29.
44
circle
.
44
.
46
.
50
.
52
.........
30. Principle of
31.
comparison
Third fundamental problem.
32.
Symmetry
33.
Further discussion
34.
Directions for discussing an equation
35.
Points of intersection
36.
Transcendental curves
Discussion of an equation
.
.
.
.
.
.
.
.
55
60
.
.
53
.65
66
.
.67
69
t
72
CHAPTER IV
THE STRAIGHT LINE AND THE GENERAL EQUATION OF THE FIRST DEGREE
37. Introduction
.
....
.
.
76
39.
The degree of the equation of a straight line
The general equation of the first degree, Ax
40.
Geometric interpretation of the solution of two equations of the
38.
first
-f
By + C =
76
.
.
80
degree
determined by two conditions
42. The equation of the straight line in terms of its slope and the coordinates of any point on the line
41. Straight lines
43.
44.
45.
46.
47.
The equation 91 the straight line in terms of its intercepts
The equation/of the straight line passing through two given points
The normal form of the equation of the straight line
...
........
The distance from a line to a point
The angle which a line makes with a second
line
The system of lines parallel to a given line
The system of lines perpendicular to a given line
The system of lines passing through the intersection
lines
.
.
86
87
88
92
96
100
107
.
.
lines
83
104
Systems of straight
49.
61.
.
.
48.
50.
77
of
.
108
two given
HO
CONTENTS
vii
CHAPTER V
THE CIRCLE AND THE EQUATION
SECTION
52.
The
53.
Circles determined
54.
general equation of the circle
Systems
a; 2
+ y 2 + Dx + Ey + F =
PAGE
...
.115
116
by three conditions
120
of circles
CHAPTER VI
POLAR COORDINATES
125
55.
Polar coordinates
56.
67.
Locus of an equation
Transformation from rectangular to polar coordinates
58.
Applications
132
59.
Equation of a locus
133
CHAPTER
126
.
.
.
130
VII
TRANSFORMATION OF COORDINATES
60. Introduction
61.
.
136
.
...
136
Translation of the axes
62. Rotation of the axes
.
63.
General transformation of coordinates
64.
Classification of loci-
65.
Simplification of equations
66. Application to
138
.
.
139
140
by transformation of coordinates
equations of the first and second degrees
.
.
CHAPTER
141
144
.
VIII
CONIC SECTIONS AND EQUATIONS OF THE SECOND DEGREE
....
68.
Equation in polar coordinates
Transformation to rectangular coordinates
69.
Simplification and discussion
70.
Simplification and discussion of the equation in rectangular coordinates. Central conies, e ^ 1
71.
Conjugate hyperbolas and asymptotes
72.
The
67.
.
.
nates.
The
parabola, e
=
1
.
.
.
.
.
149
154
of the equation in rectangular coordi.
.
equilateral hyperbola referred to
.
:
.
.154
.
158
165.
its
asymptotes
.
.
.
167
CONTENTS
viii
PAGE
SECTION
74.
Focal property of central conies
Mechanical construction of conies
75.
Types of
73.
168
....
76.
loci of equations of the second degree
Construction of the locus of an equation of the second degree
77.
Systems of conies
168
170
173
.
176
CHAPTER IX
TANGENTS AND NORMALS
........
The
slope of the tangent
Equations of tangents and normals
80. Equations of tangents and normals to the conic sections
78.
180
182
79.
81.
82.
y
83.
Tangents to a curve from a point not on the curve
Properties of tangents and normals to conies
....
Tangent
184
.
.
.
.
.
186
188
in terms of its slope
192
CHAPTER X
CARTESIAN COORDINATES IN SPACE
196
84. Cartesian coordinates
85.
198
Lengths
Orthogonal projections.
CHAPTER XI
SURFACES, CURVES, AND EQUATIONS
86. Loci in space
87. Equation of a surface.
201
First fundamental
90.
problem
Equations of a curve. First fundamental problem
Locus of one equation. Second fundamental problem
Locus of two equations. Second fundamental problem
91.
Discussion
88.
89.
.
the
.
.
.
curve.
Third fundamental
equation of a surface.
Third fundamental
equations of
a
206
problem
92. Discussion
of the
207
problem
__
205
205
.
.
201
202
.
.
.
of
.
93.
Plane and straight line
94.
The sphere
95.
Cylinders
96.
Cones
97.
Non-degenerate quadrics
210
211
.
.
.
213
.
214
.
.
.
.
.
.
.
.215
UNIVERSITY',
OF
ANALYTIC GEOMETRY
CHAPTER
I
REVIEW OF ALGEBRA AND TRIGONOMETRY
1.
Numbers. The numbers arising in carrying out the operatwo kinds, real and imaginary.
real number is a number whose square is a positive number.
tions of Algebra are of
A
Zero also
is
a real number.
A pure
tive
imaginary number is a number whose square is a neganumber. Every such number reduces to the square root of
a negative number, and hence has the form b
V
1, where b is a
2
and
number,
(V I)
An imaginary or complex number is a number which may be
written in the form a + b
1, where a and b are real numbers,
and b is not zero. Evidently the square of an imaginary number
is in general also an imaginary number, since
=1.
real
V
+ b V^T) =
2
(a
which
2.
is
imaginary
Constants.
A
if
a
is
a2
- b + 2 ab V^l,
2
not equal to zero.
quantity whose value remains unchanged
is
called a constant.
Numerical or absolute constants retain the same values in
problems, as
2,
3,
Vl,
all
TT, etc.
Arbitrary constants, or parameters, are constants to which any
one of an unlimited set of numerical values may be assigned, and
these assigned values are retained throughout the investigation.
Arbitrary constants are denoted by letters, usually by letters from the
part of the alphabet. In order to increase the number of symbols at our
first
1
ANALYTIC GEOMETRY
2
it
disposal,
example
is
convenient to use primes (accents) or subscripts or both.
For
:
Using primes,
"a prime or a
first"), of' (read "a double prime or a second"),
third"), are all different constants.
a' (read
a'" (read
"a
Using subscripts,
&!
(read
"6
"
one"), 6 a (read 6 two"), are different constants.
Using both,
c/(read "c one prime"),
c8
"
(read
"c
three double prime"), are different
constants.
3.
The
quadratic.
Typical form.
Any
quadratic equation
in the
may by transposing and collecting the terms be written
Typical
Form
Ax 2 + Bx
(1)
+ C = 0,
in which the unknown is denoted by x. The coefficients A, B, C
are arbitrary constants, and may have any values whatever,
except that A cannot equal zero, since in that case the equation
would be no longer of the second degree. C is called the con-
stant term.
The left-hand member
Ax*
(2)
+ Bx + C
called a quadratic, and any quadratic may be written in this
Typical Form, in which the letter x represents the unknown.
4 A C is called the discriminant of either (1)
The quantity B 2
is
or (2), and
That
is,
is
denoted by A.
the discriminant
A
of a quadratic or quadratic equa-
tion in the Typical Form is equal to the square of the coefficient
of the first power of the unknown diminished by four times the
product of the coefficient of the second power of the unknown
by the constant term.
The roots of a quadratic are those numbers which make the
quadratic equal to zero when substituted for the unknown.
The roots of the quadratic (2) are also said to be roots of the
quadratic equation
(1).
to satisfy that equation.
A
root of a quadratic equation
is
said
REVIEW OF ALGEBRA AND TRIGONOMETRY
In Algebra
shown that
it is
(2) or (1)
has two roots, Xi and
obtained by solving (1), namely,
(3)
Adding these
values,
we have
x
(4)
-f-
B
=
x
^Vw*
\&
Multiplying gives
XiX z
(5)
=
'A
Hence
Theorem
I.
T^e sum of the roots of a quadratic is equal to the
power of the unknown with its sign changed
coefficient of the first
divided by the coefficient of th& second power.
The product of the roots equals the constant term divided by the
coefficient of the second power.
The quadratic
+
Ax*
(6)
as
(2)
may
may be written in the form
Bx + C =*A(x - x,} (x - * 2),
be readily
member and
shown by multiplying out the right-hand
substituting from (4) and (5).
For example, since the roots of 3 a; 2
tically 3
32-43 + 1
= 3*(x - 1) (x -
The character of the
4 a;
+1=
are 1 and
,
w$ have
iden-
*).
roots x l
and x z as numbers
(
1)
when
the
C
are real numbers evidently depends entirely
A, B,
the
discriminant.
This dependence is stated in
upon
coefficients
Theorem
and
II.
If the
coefficients
of a quadratic are real numbers,
if the discriminant be denoted by A, then
4/i
*The
when A
when A
when A
=
is
is
is
positive the roots are real and unequal;
zero the roots are real and equal ;
negative the roots are imaginary.
is read "is identical with," and means that the two expressions
sign
connected by this sign differ only inform.
ANALYTIC GEOMETRY
4
In the three cases distinguished by Theorem II the quadratic
may be written in three forms in which only real numbers appear.
These are
'
C = A (xXi) (x x z ), from (6), if A is positive ;
Ax 2 + Bx + C = A(xx ) 2 from (6), if A is zero ;
Ax 2 + Bx
-f-
1
,
AC-B
\2
Ax* + Bx + C
B
= A\[-/(* + o~r)
The
proved thus
last identity is
adding and subtracting
4.
C
2>A )
L\
L
-
H
2
~]
Y if
7-7*
^
A
J
^ is negative.
:
within the parenthesis.
If one or both of the coefficients
Special quadratics.
B
and
in (1), p. 2, is zero, the quadratic is said to be special.
CASE
C
I.
Equation
=
(1)
(1)
0.
now becomes, by factoring,
Ax z + Bx = x(Ax + B) =
0.
r>
Hence the
roots are
a quadratic equation
is zero.
And
stant term
is
x1
=
0,
x2
=
Therefore one root of
yl
zero if the constant term of that equation
is a root of a quadratic, the con-
conversely, if zero
must disappear. For
we have C = 0.
if
x
=
satisfies (1), p. 2,
substitution
CASE
II.
Equation
B=
(1), p. 2,
(2)
From Theorem
(3)
0.
now becomes
Ax 2
I, p. 3,
x^
+C=
-f x =
2
0.
0,
that
!=-*
is,
by
REVIEW OF ALGEBRA AND TRIGONOMETRY
if
Therefore,
the coefficient of the
in a quadratic equation
but have opposite signs.
first
5
power of the unknown
the roots are equal numerically
Conversely, if the roots of a quadratic
is zero,
equation are numerically equal but opposite in sign, then the
coefficient of the first power of the unknown must disappear. For,
sum
since the
B=
of the roots
is zero,
CASE
B=C=
III.
Equation
have, by
Theorem
I,
(1), p. 2,
0.
now becomes
Ax 2 =
(4)
Hence the
always
0.
roots are both equal to zero, since this equation
0, the coefficient A bei ng, by hypothesis,
that x 2
requires
5.
we must
0.
=
v
different
Cases
when
from
zero.
the roots of a quadratic are not independent.
between the roots Xi and x z of the Typical
If a relation exists
:Form
Ax 9
+ Ex +
C
=
0,
then this relation imposes a condition upon the coefficients A,
B, and C, which is expressed by an equation involving these
constants.
B
2
For example, if the roots are equal, that
-4: AC = 0, by Theorem II, p. 3.
Again, if one root is zero, then x& 2 =
Theorem
p.
I,
is,
;
if Xi
hence
=x
C
2,
then
= 0,
by
3.
This correspondence
may
be stated in parallel columns thus
Quadratic in Typical
Relation between the
roots
:
Form
Equation of condition
by the
satisfied
coefficients
In many problems the coefficients involve one or more arbitrary
constants, and it is often required to find the equation of condi-
by the latter when a given relation exists between
Several examples of this kind will now be worked out.
tion satisfied
the roots.
ANALYTIC GEOMETRY
6
Ex.
1.
What must
be the value of the parameter k
if
zero
-6z + 2 -3fc-4=0?
B = - 6, C = k' - 3 k - 4. By
Case
is
a root of
the equation
2x2
(1)
Solution.
is
Here
A = 2,
fc
2
C=
a root when, and only when,
Ex.
zero
or
1.
For what values of k are the roots of the equation
2.
kx 2
real
=4
k
Solving,
I, p. 4,
0.
and equal
Solution.
+ 2kx-4x = 2-3k
?
Writing the equation in the Typical Form, we have
+
fcx2
(2)
-
k
(2
+
4)x
(3
k
-
2)
= 0.
Hence, in this case,
A = k, B = 2k-4, C = 3k-2.
Calculating the discriminant A,
A=
(2
k
-
= _8fc2
By Theorem
A=
II, p. 3,
-
2
4)
we
4k
get
k - 2)
= -8(A:2 +
(3
-8A;-f 16
&
- 2).
the roots are real and equal when, and only when,
'
k*
.-.
k=
Solving,
+
-
k
2 or
2
=
0.
Ans.
1.
Verifying by substituting these answers in the given equation
when fc=-2,
when fc= 1,
(2)
becomes
(2)
becomes
the equation
the equation
"$Hence, for these values of
formed as in
^ Ex.
a^6,
3.
fc,
x2
the left-hand
fc,
2
+1=0,
member
:
or -2(x-f 2) 2 =0;
or
(x
1)
may be
of (2)
m
(3)
trans-
equation of condition must be satisfied by the constants
the roots of the equation
if
(62
+
a2 m 2) y2
+
+ a2fc2 _
2a? kmy
a2 &2
=Q
are equal ?
Solution.
2 =0.
(7), p. 4.
What
and
2x2 -8x-8=0,
(2)
The equation
already in the Typical
(3) is
Form
;
hence
A = 6 + a m B = 2 a km, C = a?k2 - a 6
2
2
2
2 2.
2
,
By Theorem
II, p. 3,
A=
the discriminant
4 a4 fc2
m2 - 4 (62 +
A must
a2 m 2 ) (a2 &2
vanish
-
a2 62 )
Multiplying out and reducing,
a2 &2 (fc 2
-
a2 m 2
-
62 )
= 0.
Ans.
;
hence
= 0.
REVIEW OF ALGEBRA AND TRIGONOMETRY
Ex.
7
For what values of k do the common solutions of the simultaneous
4.
equations
3x + 4y =
(4)
x2
(5)
become
+
y2
fc,
= 25
identical ?
Solution.
(4) for y,
Solving
we have
= t(*-3x),
y
(6)
and arranging
Substituting in (5)
25 x
(7)
Let the roots of
2
-
6 kx +~k*
be x\ and x 2
(7)
corresponding values y\ and y z of
=
yi
(8)
and we
shall
have two
in the Typical
Then
.
y,
Form
gives
- 400 = 0.
substituting in (6) will give the
namely,
i(&
common
solutions
and
(xi, y\)
y2 )
(x 2 ,
of (4)
and
(5).
But, by the condition of the problem, these solutions must be identical.
Hence we must have
=
xi
(9)
If,
however, the
be equal.
first
X 2 and y l
of these
is
=
true (xi
y2
.
= x2 ),
then from
(8) y\
and y^
will also
Therefore the two
and only when,
criminant
A
common
of (7) vanishes
.-.
Solving,
Verification.
solutions of (4)
and
become identical when-,
(5)
the roots of the equation (7) are equal; that
(Theorem
is,
when
A;
Substituting each value of k in
(7),
when A; =25, the equation (7) becomes x 6x + 9=0, or (x 3) 2 =0
when k= -25, the equation (7) becomes x2 + 6x + 9=0, or (x-f 3) 2 =0;
;
(6),
3,
25
=
=
and x
3,
Therefore the two
common
when k
=
=
of these values of
if
k
if
k
=
=
..
x=3
.-.x=
;
3.
substituting corresponding values of k and x,
25 and x
when k
dis-.
36 2 - 100 (fc* - 400) = 0.
=
A*
625,
25.
Ans.
k = 25 or
A=
2
Then from
the
II, p. 3).
fc,
we have y = ^(25 9)=4;
we have y = $ ( 25 -f 9) =
solutions of (4)
and
(5)
4.
are identical for each
namely,
25, the
26, the
common
common
solutions reduce to x
solutions reduce to x
=
=
3,
=4
3, y =
y
;
4.
Q.E.D.
ANALYTIC GEOMETRY
8
PROBLEMS
1.
Calculate the discriminant of each of the following quadratics, deterof the roots, and write each
mine the sum, the product, and the character
quadratic in one of the forms (7), p. 4.
2x2 _ 6x + 4.
x2 - 9x - 10.
1 - x - x2
(a)
(b)
(e)
.
(c)
2.
(d)
For what
(f)
2k - 3x2 + 6x - &2 +
(b)
3.
For what
(a)
(c)
(d)
5x2 - 3x
= 0.
(f)
(g)
Ans. k
Ans. k
= l.
= - I or 3.
Verify your answers.
Ans. k
Ans. k
Ans. k
0.
+
=-f
= 6.
= ^-.
.
Ans. None.
Ans. k =
&.
Ans. None.
Ans. k=-$.
= 0.
x2 -f kx + k2 + 2 = 0.
x2 - 2 kx - k - I = 0.
(e)
4.
0.
-3x-l = 0.
x2 - kx + 9 = 0.
2 kx2 + 3 kx + 12 =
2 x2 + kx - 1 = 0.
(b)
3
real values of the parameter are the roots of the following
equations equal ?
fcx 2
parameter k will one root of each of the
real values of the
following equations be zero ?.
2
- 3 k2 + 3 =
(a) 6x '+ 5 kx
4x2 - 4x + 1.
5x2 + lOx + 5.
3x2 - 5x - 22.
5fc2
Derive the equation of condition in order that the roots of the following
may be equal.
equations
(a)
m x + 2 kmx - 2px = - k2
(b)
x2
2
(c)
5.
2
Ans. p (p - 2 km) = 0.
Ans. p (m 2p - 2 6) = 0.
Ans. fr2 = 2 a 2 m.
.
+ 2 mpx + 2 6p = 0.
2 mx 2 + 2 bx + a2 = 0.
For what
real values of the
parameter do the
common
solutions of the
following pairs of simultaneous equations become identical ?
= k, x2 + y2 = 5.
-4ns. k =
(a) x + 2 y
(b)
(c)
(d)
(e)
(f)
(g)
(h)
= mx 1, x2 = 4 y.
2 x - 3y = 6, x 2 + 2 x = 3 y.
y = mx + 10, x2 + y2 = 10.
Ix + y - 2 = 0, x2 - 8 y = 0.
x + 4 y = c, x2 + 2 y 2 = 9.
2 y = 0, x + 2 y = c.
a2 + y 2 - x
x2 + 4 y 2
8 x = 0, mx - y - 2 m = 0.
y
6. If the
are to
(a)
(b)
(c)
Ans.
-4ns.
.Ans.
5.
m=
1.
= 0.
m=
3.
6
Ans. None.
=
=
^.ns.
c
Ans.
c
^.ris.
None.
9.
or
5.
common solutions of the following pairs of simultaneous equations
become
identical,
what
is
the corresponding equation of condition
+ ay = a&, y2 = 2px.
=
mx + 6, .Ax2 + .By = 0.
y
y = m(x- a), By 2 + Dx = 0.
bx
.
?
+ ap) = 0.
B (mzB - 4 bA) =
2
- D) =
D(4 am
Ans. ap (2 62
^Ins.
-Ans.
0.
0.
REVIEW OF ALGEBRA AND TRIGONOMETRY
6.
A
Variables.
variable
is
a quantity to which.
jri
jjfr
investigation, an unlimited number of values can be assigned.
In a particular problem the variable may,< in general, assume any
value within certain limits imposed by the nature of th$ problem^
It is convenient to indicate these limits by inequalities.
For example, if the variable x can assume any value between
2 and 5, that
x must be greater* than
2 and less than 5, the simultaneous inequalities
is, if
*>-2, x<5,
are written in the more compact form
-2<z<5.
Similarly, if the conditions of the problem limit the values of the variable x to
any negative number less than or equal'to
2, and to any positive number greater
than or equal to
5,
the conditions
2 OT x
=
x
^
x<
are abbreviated to
2,
and x
>
5 or x
=
5
2 and x ^5.
Write inequalities to express that the variable
to 5 inclusive.
(a) x has any value from
2 or greater than
1.
(b) y has any value less than
8 nor greater than
(c) x has any value not less than
7.
2.
In Analytic Geometry
Equations in several variables.
we
are concerned chiefly with equations in two or more variables.
An equation is said to be satisfied by any given set of values
of the variables if the equation reduces to a numerical
equality
these values are substituted for the variables.
when
For example, x
=
y
2,
=
3 satisfy the equation
+ 32,2=35,
+ 3(- 3)2 = 35.
2x2
since
2(2)2
Similarly, x
=
1,
y
=
0, z
=
4 satisfy the equation
2z2_3 y 2 + 2;2_i 8 =
since
2
(
1)2
3
X
+
(
4)
2
18
o,
= 0.
*
The meaning of greater and less for real numbers ( 1) is defined as follows a is
when a - b is a positive number, and a is less than b when a - b is negative.
Hence any negative number is less than any positive number and if a and b are both
negative, then a is greater than b when the numerical value of a is less than the numer:
greater than b
;
ical
value of
6.
Thus 3 < 5, but - 3 > - 5.
the inequality sign.
Therefore changing signs throughout an inequality reverses
ANALYTIC GEOMETRY
10
An
equation
said to be algebraic in
is
any number of variables,
can be transformed into an equation
example
each of whose members is a sum of terms of the form ax my n z p,
for
x, y, 2, if it
where a
is
a constant and m, n, p are positive integers or zero.
Thus the equations
x*
+ x 2y 2 -
z
+2
a:
5
= 0,
are algebraic.
The equation
is
x*
+ y* =
a?
algebraic.
For, squaring,
we
get
+ 2 x$y* + y=a.
2 x*y* = a
x
a2 + x 2 + yz
2 ax
x
Transposing,
Squaring,
4 xy
Transposing,
x2
The
+ y2
2 ax
2 xy
2 ay
y.
+ 2 xy.
+ a2 = 0.
2 ay
Q.E.D.
an algebraic 'equation is equal to the highest
of
An algebraic equation is said to
its terms.*
any
with
to
the
variables when all its terms are
Ibe arranged
respect
degree of
degree of
transposed to the left-hand side and written in the order of
descending degrees.
For example, to arrange the equation
2x' 2
+ 3y' + 6x'-2x'y'-2 + z/3 = x'*if - y'*
with respect to the variables z', y',
x '3
This equation
_ yfvtf + 2 X
is
'2
we
transpose and rewrite the terms in the order
2x'y'
+ y'2 + 6a/ + 3y'
2
= 0.
of the third degree.
An equation which
is
not algebraic
is
said to be transcendental.
Examples of transcendental equations are
y
= sinx,
y
= 2*, logy=3x.
PROBLEMS
the
1. Show that each of the following equations is algebraic; arrange
terms according to the variables x, y, or x, y, z, and determine the degree.
(e)
*
The degree
of
y
= 2 + Vx2 -2x-5.
any term
is
the
sum
of the exponents of the variables in that term,
REVIEW OF ALGEBRA AND TRIGONOMETRY
+
y .= x
(f)
6
11
+ V2x2 -6x
W*=-|j>
y
(h)
2.
Show
= ^ix + B + Vix'J 4- JMTx + N.
*
homogeneous quadratic
that the
Ax2 + Bxy +
Cy*
may be written in one of the three forms below analogous to
2
4
satisfies the condition given
the discriminant A
=B
CASE
CASE
II.
CASE
III.
I.
AC
Ax + Bxy + Cy = A(x- hy] (x - y),
.Ax 2 + Bxy + Cy 2 = A(x- ky) 2 if A =
^1x2 + Bxy + Cy* = A
+
2
2
I2
,
(7), p. 4,
if
:
if
A>
;
;
', if
A<0.
[(x
Functions of an angle in a right triangle. In any right
triangle one of whose acute angles is A, the functions of A are
8.
defined as follows
sin
:
opposite
A = -**
side
A=
(/
,
/
esc
A=
-
sec
adjacent side
-=-*
,
hypotenuse
A
adjacent side'
hypotenuse
_ opposite side
tan A =
adjacent side
From
cot
the above the theorem
B
^
opposite side'
hypotenuse
cos
hypotenuse
.4
J
= adjacent
easily derived
is
side
^-.
opposite side
:
In a right triangle a side is equal to the
product of the hypotenuse and the sine of the
angle opposite to that side, or of the hypotenuse and the cosine of the angle adjacent to
that side.
A
C
b
an angle
XOA
9.
considered
is
OA
erated by the line
an
initial
positive
in
as
general.
gen-
In
Trigonometry
<
rotating from
The angle is
position OX.
when OA rotates from OX
counter-clockwise,
the
Angles
direction
of
and
negative
rotation
of
when
OA s
clockwise.
*The
coefficients A, B,
Cand the numbers
fj, I 2
are supposed real.
ANALYTIC GEOMETRY
12
The
fixed line
terminal
OX
is called
the initial line, the line
OA
the
line.
Measurement of angles. There are two important
of measuring angular magnitude, that is, there are
methods
two unit
angles.
Degree measure. The unit angle
tion, and is called a degree.
is
^J^ of a complete revolu-
The unit angle is an angle whose subtendequal to the radius of that arc, and is called a radian.
fundamental relation between the unit angles is given by
Circular measure.
ing arc
The
is
the equation
= IT radians
180 degrees
Or
also,
by solving
(IT
= 3.14159
).
this,
1 degree
=
1 radian
=
^=
loU
80
1
.0174
radians,
= 57.29
degrees.
7T
These equations enable us to change from one measurement to
In the higher mathematics circular measure is always
another.
used, and will be adopted in this book.
The generating line is conceived of as rotating around through
as
many
Any
real
conversely,
10.
we
revolutions as
number
is
any angle
is
Hence the important
choose.
result
the circular measure of some angle,
measured by a real number.
Formulas and theorems from Trigonometry.
1.
cotx
=
tanx
2.
tanx
= sinx
;
secx
;
cotx
cosx
4.
;
cscx
=
smx
= cosx
sin x
cosx
3.
=
+ cos2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = csca z.
esc x
sin (- x) = - sin x esc (- x) =
=
=
secx;
cosx;
x)
sec(
x)
cos(
cotx.
tan(- x) = tanx cot (- x) =
sin2
x
;
;
;
;
;
:
and
REVIEW OF ALGEBRA AND TRIGONOMETRY
5.
13
- x) = sinx sin (?r + x) = - sinx
- x) = - cosx
cosjtf ._&.= -<?osxj
tanx jftan (ie + x) = tanx
tan (it
x) =
sin
(if
;
cos (x
;
;
;
6.
-x)
sin(-
= cosx;
sin
(|
cos(--x)=sinx; cos^
sin (2
Tt
8.
sin (x
+
9.
sin (x
x)
= sin (
=
x)
cosx;
+ x) = - sinx;
+ x) = - cotx.
tanf-
tan(- -x)=cotx;
7.
+ x) =
O
sin x, etc.
o
y)
= sinx cosy +
cos x sin
y)
=
sin x cos y
cos x sin y.
10. cos (x
+ y} =
cos x cos y
sin x sin y.
11. cos(x
y)
=
cos x cosy
+ sinx sin y.
12
tan (x
14. sin 2 x
+
y)
--
+ tan y
= tanx
I - tanx tany
= 2 sin x cos x
;
cos 2 x
y.
- y) =
tanx
tany
+ tanx tany
.
13. tan (x
=
sin 2
cos 2 x
x
;
1
tan 2 x
=
2 tanx
x
+ cosx tan-=
;
16.
Theorem.
Law
In any triangle the sides are proportional
of sines.
to the sines of the opposite angles;
a
b
c
Lqw of cosines. In any triangle the square of a side
of the squares of the two other sides diminished by twice the
product of those sides by the cosine of their included angle ;
17.
Theorem.
equals the
that
sum
a2
is,
18.
Theorem.
=
62
+
c2
2 be cos A.
Area of a triangle. The area of any triangle equals one
two sides by the sine of their included angle
half the product of
that
is,
area
;
=
ab sin
C=
i
be sin
A = ^ ca sin B.
ANALYTIC GEOMETRY
14
11.
Natural values of trigonometric functions.
Angle in
Radians
REVIEW OF ALGEBRA AND TRIGONOMETRY
Angle in
Radians
15