Caesy Analytic Geometry

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CORNELL UNIVERSITY LIBRARY

MATHEMATICS

Cornell University Library

QA

552.C33

A treatise on the analytical geometry of

3 1924 001 520 455

The

original of this

book

is in

the Cornell University Library.

There are no known copyright

restrictions in

the United States on the use of the

text.

http://www.archive.org/details/cu31924001520455

DUBLIN UNIVERSITY PRESS SERIES.

A

TREATISE

ANALYTICAL GEOMETRY AND CONIC SECTIONS,

POINT, LINE, CIRCLE,

CONTAINING

&r %am&

of

its

mmt wmt titenmn,

WITH NUMEROUS EXAMPLES. BY

JOHN CASEY,

LL.D.,

F.R.S.,

Fellow of the Royal University of Ireland; Member of the Council of the Royal Irish A cademy ; Member of the Mathematical Societies of London and France ; and Professor of the Higher Mathematics and Mathematical Physics in the Catholic University of Ireland.

DUBLIN: HODGES, FIGGIS, &

LONDON LONGMANS, GREEN, & :

CO.,

CO.,

GRAFTQN-STREET.

PATERNOSTER-ROW.

1885. i

\All rights reserved.']

1

DUBLIN

:

PRINTED AT THE UNIVERSITY

PRESS,

BY PONSONBY AND WELDRICK.

;

PREFACE,

TN

the present

Work

I

have endeavoured, without exceed-

ing the usual size of an Elementary Treatise, to give a

comprehensive account of the Analytical Geometry of the

Conic Sections, including the most recent additions to the Science.

For several years Analytical Geometry has been study,

and some of the investigations

portions of this Treatise were

by myself.

These include

:

first

in the

my

special

more advanced

published in Papers written

finding the Equation of a Circle

touching Three Circles ; of a Conic touching Three Conies extending the Equations of Circles inscribed and circumscribed to Triangles to Circles inscribed and circumscribed to

Polygons of any number of sides

of the properties of circles

;

the extension to Conies

cutting orthogonally; proving

that the Tact-invariant of two Conies

Anharmonic Ratios; and some

others.

is

the product of Six

:

Preface.

iv

Of

the Propositions in the other parts of the Treatise,

the proofs given will be found to be not only simple and elementary, but in

In compiling

some instances

my Work

various authors.

have consulted the writings of

I

Those to

original.

whom

I

am most

Salmon, Chasles, and Clebsch, from the

indebted are

last

of

whom

I

have taken the comparison of Point and Line and Line Co-ordinates (Chapter

II.,

Section III.)

the

and Aronhold's

now published

notation (Chapter VIII., Section III.), first

;

for

time in an English Treatise on Conic Sections.

For Recent Geometry, the writings of Brocard, Neuberg, Lemoine, M'Cay, and Tucker.

The

Those placed

exercises are very numerous.

after the

Propositions are for the most part of an elementary character,

to

and are intended as applications of the propositions

which they are appended.

the chapters are more

The

Some have been

difficult.

from the Examination Papers

exercises at the ends of

selected

set at the Universities,

from

Roberts' examples on Analytic Geometry, and Wolstenhqlme's

Mathematical Problems. large

number

I

am

Some

are original

indebted to

Professors

Neuberg, R. Curtis,

Messrs.

and F. Purser.

J.

The work was

my

s.j.,

late

and

for a very

Crofton, and the

read in manuscript by

esteemed friend, the

;

Mathematical friends

my

lamented and

Rev. Professor Townsend, f.r.s.

by Dr. Hart, Vice-Provost of Trinity College, Dublin

;

;

and

v

Preface. Professor B. Williamson, f.r.s.

Their valuable suggestions

have been incorporated. In conclusion, I have to return

my

best thanks to the

last-named gentleman for his kindness in reading the proof sheets,

and to the Committee of the

Press Series'

'

Dublin University

for defraying the expense of publication.

JOHN CASEY. 86,

South Circular Road, Dublin, October

5,

1885.

;

[The following Course, omitting the Articles marked with

recommended Chapter VII.]

II.,

for

Junior readers

:

Chapter

I.,

Sections

asterisks, is

I.,

II., III.

Section I.;] Chapter III., Section I.; Chapters V., VI.,

——

CONTENTS, CHAPTER

I.

THE POINT. Section

I.

Cartesian Co-ordinates. page

Definitions,

...

.

.

Distance between two points,

Condition that three points

Area of a triangle Area of polygon

.

.

may be

collinear,

4

in terms of the co-ordinates of its vertices,

„

i

3 6

.

7

t ,

.

,,

Co-ordinates of a point dividing the join of two given points in

a given

Mean

ratio,

centre of any

.

number of given

Section Polar co-ordinates,

.

,

Section

Sum

of,

9

.

10

.

12

III.

....

.

Complex Variables

Section IV. Definition

.

II.

.

Transformation of co-ordinates,

8

.

points,

and mode of representation,

14

.

or difference of two complex variables,

Product of two complex variables, „ „ Examples on complex variables, Quotient

Miscellaneous Exercises,

.

.

.

.

.

.

15 ,

.

16 16

.

16 17

——

—

Contents.

viii

CHAPTER

II.

THE RIGHT LINE. Section

Cartesian Co-ordinates.

I.

fag

To

represent a right line

by an equation,

Standard form of equation,

.

]

...

.

.

i

Line parallel to one of the axes,

5

Comparison of different forms of equation,

'

To

find the angle

between two

''

lines,

Length of perpendicular from a given point on a given Equation of a line passing through two given points,

line,

.

'
,

c

.

.

centre of .

I,

prove

*

7

.

Miscellaneous Exercises.

1

Miscellaneous Exercises. 1.

Show

that the polar co-ordinates

represent the 2.

same

(p, 9)

;

(-

p, it

+

a

+ i|,

S

(>,

;

-

9

ir),

all

Prove that the three points (a,

b)

+ 28^2,

(a

;

+

6

28 v^)

;

(

form a right-angled

- &),

V2

\ 3.

9)

point.

V2y

triangle.

Find the perimeter of the quadrilateral whose vertices, taken in order,

are

(-SV3 ,s);

aVi);

(a,

4. If the three sides

(~c, -c\fl);

l:-m, m: — n, n:—l, prove that the 5.

If (x, y) (x', y')

three points of section are collinear.

be the co-ordinates of a point referred respectively to

common

rectangular and oblique axes having a axes of the

first

prove that

origin,

if

the

system bisect the angles between those of the second,

x=

[x

6. If the points (ab), 7.

-d).

(atVj,

of a triangle, taken in order, be divided in the ratios

+ y)

y=

cos -,

— a',

(a

(a' b'),

*

—y)

(x

-

be

b')

sin -.

collinear,

prove ab'=a'b.

If the co-ordinates {x'y'), (x"y"), (x"'y'") of three variable points

satisfy the relations

(x'-x") = \(x" - *"')-/*(/' -/"), {y'-y") where A and

p.

= *{y"-y'")

+ /*(#" -*"'),

p ove that the

are constants,

triangle of

which these points

are vertices is given in species. 8. If

two systems of co-ordinates have the same

origin and the

same axis

of x, prove that

x= 9.

x\y

sin

,

(oj

— to)

^

sin

Prove that the orthocentre of a triangle

gular points for the multiples tan

n10. For what mean 11

,

system of multiples

its sides,

centre of its

^

,

U*

ft

*"— .

V

triangle, a, b, c the lengths' of

prove that the co-ordinates of the centre of its inscribed

+ bx" + a~+b +

12. If

mean

the circumcentre of a triangle the

x"y" , x'"y'" be the vertices of a

ax' \

is

the

is

•

tan/?, tan.

(45)

(af\ 2af).

Ans. ,5°.

(a?

- j8) }

(*.£) Ans.

.

=

(a

(43)

sin a

cos a

4

from the following pairs of

loci of points equally distant

:

= a(t+f)(t2 + f + ';

Or thus axis of

x

1

denotes (Art. 16) a line parallel to the and x - x = o a line parallel to

at the distance y';

oiy

the axis

= o passes through the power of x'y' with respect to it is zero.

y - y' = o

:

m(x^ x )

Iinej/ -j>'-

1

at the distance x".

Hence,

y -y' -m{x -x

1

=

)

Cor. 2,

o

(65)

denotes a line passing through their intersection, that through the point x'y'. Cor. 4. «S"= o,

—In

S'=

the same manner it may be shown that if be the equations of any two loci (such as a line or two circles, &c), IS + mS' = o will denote

o,

and a circle, some curve passing through S and S'. 20.

is,

To find

the equation

all

of a

the points of intersection of

passing through two points

line

x'y 1, x"y".

Take any

variable point

points xy, x'y

1 ,

x"y" are x,

y,

1,

x

y,

1,

y",

1,

1 ,

*",

which It

xy on the

in the

first

2

we

otherwise seen that this

degree; hence

substitute x'y' for

it is

is

1

satisfy

it,

and the

the equation of a line i°.

It contains

x andj/

the equation of a right line.

xy the determinant

rows alike, and therefore will vanish x'y

(12),

the required equation.

is

may be

If

then the three

(66)

passing through the two given points.

.

line,

Hence, equation

collinear.

line passes

;

will

have two

hence the co-ordinates

through

x'y'.

Similarly

it

The Right Line.

30

The determinant

passes through x"y".

(y ~y") x-{x>- x")y from which we

(66)

expanded

+ x'y" -x"y' = o;

gives

(67)

infer the following practical rule for writing

•down the equation of a line passing through two given points

x'y',

x"y"

:

—

Place the co-ordinates of one of the given points

under

give the

of

of the other, as in the margin; then the of the ordinates of the given points will

those

'difference

coefficient

of x :

y',

the corresponding difference

sign changed will be the

the abscissa with

x',

x", y",

Lastly, the determinant, with two rows

of y.

coefficient

formed by

the given

co-

ordinates, will be the absolute term.

Cor.

1.

— If the

equation of the line joining x'y', x"y" be

Ax + By + C = o, we

written in the form

y -y" = A, Cor. 2.

— Hence

(x'~ x") =

-B,

have

x'y" - j/>' = C.

may be inferred the condition may subtend a right angle at

points x"y", x"'y"'

that the x'y'.

For, let the join of the points x'y',

x"y" be

Ax + By + C =

o,

and the join of the points x'y', x'"y'"

be A'x

+B'y+C'=o;

and, since these are at right angles to each other,

AA'+BB'=o; and, substituting, we get

(^ -

x") {x' - x"') +

(y -y"){y -y">) =

0.

(Comp.

(8).)

—— Cartesian Co-ordinates.

31

Examples. 1.

Find the equation of the join of

(2,

—

—

4), (3,

5).

+y + 2 =

Ans. x 2.

Find the medians of the triangle whose vertices are Ans. (y" +y'"

— 2y')x—(x" + x'"-2x')y + (x" + x"')y' -{y"+y'")x' =

3.

&c.

(68)

Ans. cos%(").

;

+ 4>")-

o,

pairs of points

(rcostp", rsinQ").

(rcos^/, J-sin^');

Ans. cosf

= cos£O'-$>").

+sin£(0' + 4>")^

+ j8), Jsin(a+£)};

{a cos (a

-

j8),

.

a

4

{afi,

.

5°.

2at); (at12 , 2at).

(asec^, Standi);

Ans. cos I 6".

(70)

&sin(a-j8)}.

x y Ans. cos a- +S11107 = _,

Ans..

cosfi.

b

2x-(t+t')y+2aif = o.

(71)

"

'

(72)

(asec', bta.n