An Introduction to Algebraical Geometry

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OXFORD UNIVERSITY PRESS AMEN HOUSE,

E.C,

4

London Edinburgh Glasgow New York Toronto Melbourne Capetown Bombay Calcutta Madras

HUMPHREY MILFORD PUBLISHER TO THE UNIVERSITY

PIRST EDITION IQI2 REPRINTED PHOTOGRAPHICALLY IN GREAT BRITAIN IN 1937 BY LOWE AND BRYDONE, PRINTERS, LONDON, FROM CORRECTED SHEETS OF THE FIRST EDITION

PREFACE THE

aim has been

book suitable to the beginner who wishes to acquire a sound knowledge of the more elementary parts of the subject, and also sufficient for the The syllabus for candidate for a mathematical scholarship. author's

Honour Moderations

at

to produce a

Oxford has been taken as a maximum

limit.

The main

principles observed in its construction are (1) to make of to the the student (2) knowledge :

utilize the previous

;

subject self-dependent; (3) to arrange the bookwork in such (4) to logical order that any portion can be readily found illustrate difficulties by worked-out problems, each selected with ;

a definite object (5) to graduate the exercises and to select only those which can be done by the preceding bookwork. ;

The

solutions of illustrative examples are not always the most elegant possible the probable capacity of the student at each ;

stage has been carefully considered.

During twelve

years' daily experience of teaching this subject the author has noted the difficulties common to students. For example, the average student has no idea of the form of '

an equation

'

thus, asked to find the equation of a line through a given point perpendicular to a he begins, Let gpven line, = mx + c be the line, then its "m"', &c., whereas, if he had y a clear understanding of form, he would readily write such an :

c

equation down. In order to give the -reader confidence in analytical methods, familiar properties are used as illustrations and well-known

wherever they arise naturally out of the analysis. Thus the circle is fully dealt with; methods and ideas are A large number of thereby illustrated earlier than usual. facts are noted

exercises are given in this part of the work so that the pupil can make the foundations sure the reader with special mathe;

A2

PREFACE

4

matical ability can omit many of these. Some of the work on the circle, especially that dealing with the circles of the triangle,

the author believes, new. tried to avoid obtaining analytical results by quoting geometrical results with which the reader may be This process makes some acquainted in Geometrical Conies. is,

The author has

pupils lose confidence in Analytical Geometry; others welcome it as a dodge enabling them to avoid a real understanding of in either case the result is bad. the principles of the subject All properties of the conic are developed by analytical processes Thus, for instance, the equation of following on definitions. ;

the axes of the coordinates,

is

general conic, either in Cartesian or Areal obtained from the simple definition of an axis

as a straight line about which the conic is symmetrical the This equation foci are subsequently shown to lie on the axes. :

deduced from those giving the foci by making the statement that the foci lie on the axes, a statemejit which, most probably, the reader would fail to justify except by an appeal to Geometrical Conies. Briefly, this book attempts to answer the question, What do the general equations of the first and second degree represent ?^ of the axes

is

not

'

1

What

equations represent certain known curves ? The chapter on the circle, however, comes before the general discussion of the equation of the second degree; the purpose rather than,

(

'

being to familiarize the student with the work before the more serious attack, and to cater for those examinations their syllabus to the line and circle.

A few details may be noted

:

abridged notation

is

which limit

insisted

upon

as probably the best introduction to quite general coordinates.

author's treatment of the parabola Vax + \/6//= 1 is original and will, he hopes, commend itself to teachers who have realized the difficulties boys find with the usual work. Parametric co-

The

ordinates are given their rightful prominence. In the first draft of this book point and line coordinates were treated concurrently :

convinced, however, of the relative importance of the former, the author changed his scheme it is hoped that the introductory :

chapter on line coordinates will prove useful. Special care has been given to the introduction of imaginary points, points at at infinity. The last chapter, is devoted infinity, and the line

PREFACE

5

and here tangential equations are freely of the many proofs given are new. author's first and unlimited thanks are to Mr. A. E. Jollifte,

to Areal coordinates,

used

;

The

M.A., Fellow and Tutor of Corpus Christi College, Oxford whenever a difficulty, either of arrangement or of method, has arisen he has given most helpful advice, and it is largely due to his aid :

and encouragement that

this work has been completed. read through and thoroughly criticized both the manuscript and the proofs. The author is entirely responsible for the form and accuracy of the work, but it is right to state that Mr. Jolliffe most generously placed a quantity of his own

Mr.

Jolliffe also

work

at the author's disposal thus the practical methods of drawing conies and some of the best paragraphs in the later ;

chapters are adapted from his manuscript. Chapter IV was submitted this part inserted at his suggestion, and he kindly of the work to other mathematical authorities for their criticism.

H. Wykes, M.A., spent much time and care reading the manuscript, and his suggestions were often adopted. Miss Isabella Thwaites, scholar of Girton College, Cambridge, and Mr. W. E. Paterson, M.A., have kindly read the proofs. The author is glad also to recognize the unfailing courtesy and kindness extended to him by the Clarendon Press. The author hopes he has produced a book that will not only Mr.

P.

make

the subject interesting to schoolboys, but will be a valuable companion to which later on the undergraduate will often refer

and from which he

will not readily part.

A. C.

BRADFORD, 1912.

J.

CONTENTS PAGE

CHAPTER

I

THE POINT

9

CHAPTER THE EQUATION OF THE FIRST DEGREE

CHAPTER EQUATIONS OF HIGHER DEGREES.

II

.....

28

III

CHANGE OF AXES

.

84

.

CHAPTER IV ANALYTICAL NOTATION, A REVISION AND EXTENSION

.116

.

CHAPTER V THE CIRCLE

125

CHAPTER THE GENERAL EQUATION

VI

OF THE SECOND DEGREE

CHAPTER

.

.

.

VII

THE PARABOLA

257

CHAPTER CENTRAL CONICS.

222

VIII

THE ELLIPSE AND THE HYPERBOLA

.

.

303

CHAPTER IX 378

POLAR COORDINATES

CHAPTER X LINE COORDINATES AND TANGENTIAL EQUATIONS

.

.

.

390

CHAPTER XI MISCELLANEOUS THEOREMS

.......

CHAPTER

430

XII

TRILINEAR AND AREAL COORDINATES

452

ANSWERS

527

INDEX

..........

545

CHAPTER

1

THE POINT THE method

1.

processes (a)

(I)

of algebraical analysis involves three distinct

:

The conditions

problem are represented by and algebraical expressions equations. The processes of algebra are applied to these expressions and of a geometrical

equations to obtain (c)

new

results.

These new results are translated back into geometrical guage. object of the

The

perform readily the

bookwork given

first

is

lan-

to enable the student to

and third operations.

The tendency

in this

lose sight of the geometrical significance: the student subject should take the greatest pains to acquire the habit of connecting is to

every algebraical detail with its geometrical interpretation. One of the most elementary relations between Geometry and

The number the expression for the area of a rectangle. of square units in the area of a rectangle, whose sides are a and b units of length respectively, is the product ab. The algebraical ex-

Algebra

is

pression ab may thus be said to represent the geometrical quantity, the area of a rectangle.

Algebraical proofs of the propositions in Euclid, Bk. II, are based idea. The logic of the process is here illustrated.

on this

j

// a

whole ivith

straight line be divided into

line is

equal

to the

sum

any two parts

of the squares

the

square on the

on the two parts, together

twice the rectangle contained by the tivo parts.

THE POINT

10

AB

is the straight line divided at C, let AC, CB be a and b units of length respectively. is represented algebraically by (a) The area of the square on the product (a + b) x (a + b).

If

AB

(6)

Now

(c)

But

(

=

a + 6)x(a + 6)

(a

+ 6) 2

this result represents the square

the rectangle contained by the square on is equal to

AB

Hence The algebraical *

tion

thus

;

on

A C and

CB + twice

if

',

AC+ the

square on

CB.

&c.

idea of sign is represented geometrically is the length of CB measured from C to

by

B

B

length measured from

to

A

C is

A

&_--C

b

b (")

(i)

Thus in Fig. (i) AB is of length (a + 6), in The absence of this idea in Euclid accounts

Fig.

of length (a b). of pairs

(ii)

number

for the

of propositions which are algebraically equivalent II. 12 and II. 13. II. 5 and II. 6 II. 7 ;

the

b.

B

C

direc-

+ 6,

:

e. g.

II.

4 and

;

For instance, in the example given above, if Using Fig. (ii) proposition II. 7 is derived. (a) The square on AB is represented by (a~b) (a = a 2 + b*-2ab. (b) (a~b)~ (c) This represents the sum of the squares on

b

were negative *

7>).

AC

and

BC

less

AC

twice tho rectangle contained by and BC. The drawing of graphs is a further useful step towards connecting the subjects of Geometry and Algebra. Graph drawing is now taught in all schools, of the process. 2.

and we assume that the reader has some knowledge

Cartesian Coordinates.

Rectangular and Oblique Axes.

P

The

in a plane is indicated by its distances position of a point measured in fixed directions from two chosen intersecting straight lines Ox, Oy (the coordinate axes or axes of reference) ; the axes are called rectangular

oblique.

In both

the lengths

when

Oy are perpendicular, otherwise PN be drawn parallel to Oy (the axis of y), called the x and y coordinates. Thus, if Ox,

cases, if

ON,NP

ON = h and NP =

ft,

are

P is

the point

(h, k).

The same convention with regard nometry: lines

made as in trigomeasured upwards (NP) are positive, downwards (N'P') to sign is

THE POINT

11

and again

negative,

to the right (ON) positive, to the left (ON') a and the point IJ/ is ( a, 1) where ON'

negative:

e.

N'P'~

The angle xOy

I.

g.

usually called

is

to.

P(h,k)

N

1

N

The

its ordinate. a,

b

'

is called

^-coordinate of a point

as

'

We

shall refer to

the point P(a,

l>)

'

its abscissa,

the point

P

the ^/-coordinate

whose coordinates

are

'.

Note. In the majority of pioblems it is more convenient to use rectangular axes, especially when the lengths of lines or the magnitude of for angles (i. e. metrical properties) are involved, because the expressions these quantities are much simpler when w however, that the solution of a problem

is

a right angle

:

it

may happen,

much

simplified in other due to the, more inconvenience that the axes use of the oblique by

ways clumsy formulae

is

so

is outweighed consequently the student should make himself familiar with the formulae in the more general case. It may be well to note here that as a rule the student has free choice of axes of ;

the first step is to decide what lines in the reference in any problem axes of reference, the only restriction convenient the most will make figure a variable line must not be chosen for an that they must be fixed ;

being axis nor a variable point for origin. ;

Polar Coordinates.

3.

Ofpok)

P

The (i) (ii)

is indicated by position of a point a fixed point 0, called the pole, from its distance OP (r)

the angle

(0)

axis OZ.

which

OP (the

radius vector) makes with a fixed

THE POINT

12

When

the position of P, the radius vector

is positive, to find

in a direction from the position OZ and revolve about the of a clock the to that of the hands opposite through angle if distance r, if positive, is measured along this radius from It should be noted that with negative, in the opposite direction.

must

start

;

this convention of sign the points indicated (

TT

r,

4.

+ 0),

by

(r,

0),

(

?rH-0),

r,

27r) are identical.

(ry

The polar coordinates of a point referred

to

the

line

OZ

are

connected by simple relations with the Cartesian coordinates referred to rectangular axes through 0, along and perpendicular to OZ.

(xy

Let

P be

then

the point x

Conversely

(r,

0)

= ON = r = v/a^-f

or

(,r,

r cos

T/'

;

y) ;

:

y

= ^VP =

=

and

r sin -

tan" 1

3x

Thus with these #

=

lines of reference the graphs of

2?/

1

and

r cos

=

2r sin

1

are identical.

Examples 1.

I

a.

With rectangular axes mark the positions of the points (2, 1*5), (0, 3), -3) (-2, 4-5). Note graphically that they are collinear. What

(4, 0), (8,

equation do they all satisfy ? 2. With axes inclined at 12(T, note the positions of the points (6, 1),

(-2, -3),

(5, 4), (2,

-I) (-8,

Which of these points are collinear ? 3. Mark the positions of the following (.

and

M

(0, 0),

4).

(a, fir), (a,

points

:

-Jir), (-a,

- jrr),

find their Cartesian coordinates referred to rectangular axes

through

the pole, one of which coincides with the polar axis. 4. Find the polar coordinates, referred to Ox, of the following points, whose Cartesian coordinates referred to rectangular axes are (3, 4), (5, 5), (2

A

:

2)(-2v/3,

2),

(-2-A

~2), (2V3, -2).

THE POINT The make

18

sides of a square are 4 units long choose coordinate axes so as the coordinates of the corners as symmetrical as possible, and state their coordinates. 5.

to

6.

The

;

sides of a parallelogram are 4 and 6 units of length its corners referred to suitable axes.

;

find the

;

find the

coordinates of

7. The diagonals of a rhombus are 2 and 4 units of length coordinates of its corners referred to suitable rectangular axes.

8.

Draw

9.

(vi)

Draw graphs

= JTT,

(iv)

(v)

=

x+y of

x+a

=

0,

=

0,

x

(i)

=

cosec B

are

6.

To

=

a2

(ii)

;

tan

(a, CX), its

the pole,

t/,

(iv)

Cartesian coordinates

the #-axis

making an

prove

:

y

y*

=

0.

to rectangular axes through angle 30 with the polar axis, are (x y)

x1 4

x

=

referred

(i)

3, (iii)

=

The polar coordinates of a point

10.

=

y

(ii)

0,

(i)

2r+ 5

= 4,

referred to (a) rectangular, (b) oblique axes. r cos & +2 0, (iii) 3r-sec 1, (ii) r sin

the graphs of

4#,

y

(v)

a =

(y 1/8

+ x)/(x*/$ - y}.

find the distance between two points whose coordinates are

given. I.

Cartesian Coordinates, Oblique axes.

M

Let the points be P (;r t to 0>/, and QL parallel to

LPLQ ==

then

y^ and Q

,

TT-