The elements of coordinate geometry

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uTd-nu
y:

/o-

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—

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BY THE SAME AUTHOR.
A

Treatise on Elementary Dynamics.
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THE ELEMENTS
OP

COOEDINATE GEOMETRY.

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THE ELEMENTS
OF

COOEDINATE aEOMETRY

BY

S.

L.

LONEY,

M.A.,

LATE FELLOW OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE,
PROFESSOR AT THE ROYAL HOLLOWAY COLLEGE.

^^^l^^mU. MASS.
MATH, DEPTi

MACMILLAN AND
AND NEW YOEK.
1895
[All Bights reserved.']

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CO.


CTambrtlJse:

PRINTED BY

J.

&

C.

F.

CLAY,

AT THE UNIVEBSITY PRESS.

150553
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PKEFACE.
"TN

the following work I have tried to present the

elements of Coordinate Geometry in a manner
suitable

present

Junior Students.

The

book only deals with Cartesian and

Polar

Beginners and

for

Within these

Coordinates.

that the book

is fairly

limits I venture to

complete, and that no proposi-

tions of very great importance have

The Straight Line and

more

fully

since

it

is

hope

been omitted.

Circle have been treated

than the other portions of the subject,
generally in the

elementary conceptions

that beginners find great difficulties.

There are a large number of Examples, over 1100
in

all,

and they

character.

are,

in

The examples

general, of an
are especially

the earlier parts of the book.

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elementary

numerous in


PREFACE.

vi

for

reading

portions of the proof sheets, but especially to

Mr W.

I

J.

am much

indebted to several friends

Dobbs, M.A. who has kindly read the whole of the

book and made many valuable suggestions.
For any
shall

be

criticisms,

suggestions, or

grateful.
S.

EoTAIi

corrections, I

HOLLOWAY COLLEGE,

Egham, Surbey.
July

4, 1895.

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L.

LONEY.


CONTENTS.
CHAP.
I.

II.

Introduction.

...

Algebraic Kesults

Coordinates. Lengths of Straight Lines and
Areas of Triangles
Polar Coordinates

Equation to a Locus

Locus.

IV.

The Straight Line. Eect angular Coordinates

....
.

.

.

pendicular

.

.

.

Length of a perpendicular

.

.

.

Line.

.

.

51

.

.

66

.

.

.

loci

Equations representing two or more Straight
Lines
lines given

by one equation

Greneral equation of the second degree

VII.

42

Polar Equations and

Equations involving an arbitrary constant

Angle between two

39

58

Oblique Coordinates

VI.

31

.44

.

.

"

.

Bisectors of angles

Examples of

8

24

Straight line through two points
Angle between two given straight lines
Conditions that they may be parallel and per-

The Straight

1

19

III.

V

PAGE

Transformation of Coordinates
Invariants

.

.

.

.

.

73
80

88
90
94
109
115

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.

CONTENTS.

Vlii

PAGE

CHAP.

VIII.

The Circle

118

Equation to a tangent

126
137

Pole and polar

Equation to a circle in polar coordinates
Equation referred to oblique axes
Equations in terms of one variable
.

.150

.

160

Systems of Circles
Orthogonal circles

X.

148

.

.

.

IX.

.145

.

.

.

.

,

.

.160

.

Kadical axis

161

Coaxal circles

166

The Parabola

Conic Sections.

174

.

180

Equation to a tangent

Some

properties of the parabola

.

187

.

.

190

Pole and polar

195

Diameters
Equations in terms of one variable
XI.

The Parabola

{continued')

.

.198

.

....

Loci connected with the parabola
Three normals passing through a given point
Parabola referred to two tangents as axes
.

.

.

211

The Ellipse

225

Auxiliary circle and eccentric angle

Equation to a tangent

Some

Conjugate diameters

.231

.

.

....

237

......

249

.

.

properties of the ellipse

Pole and polar

Pour normals through any point
Examples of loci
XIII.

206

.217

.

XII.

206

.

.

.

.

.

.

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254
265

266

The Hyperbola
Asymptotes
Equation referred to the asymptotes as axes
One variable. Examples

242

271

284
.

296
299


.

CONTENTS.
CHAP.

XIV.

Polar Equation to, a Conic

IX

....

Polar equation to a tangent, polar, and normal

XV.

General Equation.

Tracing of Curves

Particular cases of conic sections

PAGE

306
313

,

322

....
.

322

Transformation of equation to centre as origin
Equation to asymptotes
Tracing a parabola

......
......
......

Tracing a central conic

.

.

.

.

.

.

Eccentricity and foci of general conic

XVI.

General Equation
Tangent
Conjugate diameters

326

.

329
332
338
342

349
349
352

Conies through the intersections of two conies

356

The equation S=Xuv

358

......
.......

General equation to the

j)air

of tangents

drawn

from any point

The director
The foci
The axes

circle

367
369

Lengths of straight lines drawn in given directions
to meet the conic
Conies passing through four 23oints
.

.

.

Conies touching four lines

The

LM=B?

conic

XVII. Miscellaneous Propositions

On

364
365

....

370
378
380
382
385

the four normals from any point to a central
conic

Confocal conies

.......

Circles of curvature

and contact of the third order

385

.

Envelopes

Answers

392
398
407

i

.

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—

xiii


ERKATA.
Page
„
,,

,,

87,

Ex.

27, line 4.

235, Ex. 18, line 3.

„

,,

282, Ex.

,,

3.

line 5.

For "JR" read " S."

For "odd" read "even."
Dele

"and Page

37,

Ex. 15."

For "transverse" read "conjugate."

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CHAPTER

I.

INTRODUCTION.

SOME ALGEBRAIC RESULTS.
1.

Quadratic Equations.

The

roots of the quad-

ratic equation
a'3^

may

easily

+ 6x + c =

be shewn to be

-

&

+

•JlP'

— 4ac

1

-b- s/b^ — 4:aG

'^"'^

•

2^

2i.

They are

and unequal,

equal, or imaginary,
according as the quantity b^—iac is positive, zero, or negative,

therefore real

i.e.

2.

and

according as

b^

=

4:ac.

Relations between the roots of any algebraic equation
of the terms of the equation.

the coejicients

If any equation be written so that the coefficient of the
highest term is unity, it is shewn in any treatise on Algebra
that

the sum of the roots is equal to the coefficient of
(1)
the second term with its sign changed,
(2)

the

at a time,

is

sum of the products of the roots, taken two
equal to the coefficient of the third term,

the sum of their products, taken three at a time,
(3)
equal to the coefficient of the fourth term with its sign
changed,
is

and so
L.

on.

e
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1


,

.

COORDINATE GEOMETRY.
Ex.

If

1.

a and

/3

ax'^

2.

If a,

b

+ bx + c = 0,

i.e.

x^

c

+ - x + ~ = 0,
a

—
b

a + p=

we have

Ex.

be the roots of the equation

a

c

^

and a^ = -.

and 7 be the roots of the cubic equation

j8,

ax^ + bx^ + cx + d=0,
i.e.

x^+-x^

of

a

we have

+-x + - = 0,
a

a

+ p + y:

a

^y + ya + a^=:and

o-Pl-

3.

shewn that the solution

It can easily be

of the

equations

+

a^x

h^y

+

G^z

= 0,

a^ + h^y + c^z = 0,

and

X
IS

y

~ ^2^1

^1^2

^1^2 ~ ^2^1

'^1^2

~ ^2^1

Determinant Notation.
4.

The

is called

quantity-

a determinant of the

second order and stands for the quantity

Exs.

(1)

d-yf

d^

^1,

h

a-})^

— aj)^,

so that

= Ob^^ — 6»2&i

\%^\ = 2x5-4x3 = 10-12=-2;

;'

|

!4, 5i

3,
(ii)

-7,

-4|

-6 = 3

X

(

-

6)

-{-

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7)

X

(

- 4) = 18 - 28 = -

10.


DETERMINANTS.
«!,

5.

The quantity

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COORDINATE GEOMETRY.

8.

The quantity
j

(h.1

^2>

%J

^4

61,

&2)

hi

h

^11

^25

^1) ^2

^3>
5

^3) ^4

called a determinant of the fourth order and stands for
the quantity

is

K
«i X

^2»

<^2>

h,
^3

^4

J

— Clo

^3> ^4

X

i^lJ

^35

C-,

^3}

\

1

1

&i,

+ 6^3

h
4

33

5

62J

X

^4!
C^

1

?

2

5

4

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&1,
cCj_

X


.

ELIMINATION.

5

Elimination.
Suppose

11.

we have

the two equations

+ a^y =

aj^x

(1),

\x +b^y ^0
two unknown quantities x and

between the
be some relation holding between the four
^or, from (1), we have
bi, and 63

(2),

y.

There must

coefficients

6*i, ctaj

•

and, from

(2),

we have

y~

%'

-=—
y

K

Equating these two values

i.e.

=-^

X

of -

we have

— ajb^ =

a-J)^

(3).

The result (3) is the condition that both the equations
and (2) should be true for the same values of x and y.
The process of finding this condition is called the eliminating of X and y from the equations (1) and (2), and the
result (3) is often called the eliminant of (1) and (2).
Using the notation of Art. 4, the result (3) may be
(1)

written in the form

1

)

'^

0.

is obtained from (1) and (2) by taking the
x and y in the order in which they occur in
the equations, placing them in this order to form a determinant, and equating it to zero.

This result

coefficients of

12.

we have
a-^x + a^y + a^^ =
\x+ h^y^ h^z =
G^x + G^y + C3S =
unknown quantities

Suppose, again, that

and
between the three

the three equations

(1),

(2),

(3),
x, y,

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and

z.


COORDINATE GEOMETRY.

6

By

dividing each equation by z

we have

three equations

X

between the two unknown quantities — and y

z

%,

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z

Two

of


ELIMINATION.
14.

and

If again

we have

the four equations

a-^x

+

dil/

+

cf'zZ

+ a^u =

0,

h^x

+

h^y

+

b^z

+

b^u

=

0,

Ci«;

+

c^i/

+

G^z

+

c^u

=

0,

djX + d^y

+

d.^z

+

d^ — 0,

could be shewn that the result of eliminating the four
quantities cc, y, z^ and u is the determinant

it

«1J

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CHAPTER
COORDINATES.

II.

LENGTHS OF STRAIGHT LINES AND
AREAS OF TRIANGLES.

OX

07

and
15. Coordinates. Let
be two fixed
The line
is
straight lines in the plane of the paper.
the axis of y, whilst the
called the axis of cc, the line
two together are called the axes of coordinates.

OX

OY

The point

is

called the origin of coordinates or,

more

shortly, the origin.

F

From any point in the
plane draw a straight line

OF to meet OX
M.
The distance OM is called
the Abscissa, and the distance

parallel to

in

MP the Ordinate of the point
P, whilst the abscissa and the
ordinate together are called
its Coordinates.

OX

Distances measured parallel to
or without a suffix, {e.g.Xj, x.-^... x\
measured parallel to OY are called
suffix, (e.g.

2/i, 2/2,---

If the distances

the coordinates of

2/'.

y",---)-

OM and MP

P are,

are called a?, with
x",...), and distances
y, with or without a

be respectively x and ?/,
by the symbol

for brevity, denoted

{x, y).

Conversely, when we are given that the coordinates of
For from
we
a point
are (x, y) we know its position.
{—x) along
and
have only to measure a distance

P

OM

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OX


COORDINATES.
then from 21 measure a distance

9

MP

OY

{=y) parallel to
arrive at the position of the point P.

For example
be equal to the unit of length and
in the figure, if

and we

MP= WM,

OM

P is the point (1, 2).
Produce XO backwards to form
then

the line OX' and
backwards to become OY'. In Analytical Geometry
we have the same rule as to signs that the student has
already met with in Trigonometry.
Lines measured parallel to OX are positive whilst those
measured parallel to OX' are negative ; lines measured
parallel to OY are positive and those parallel to OY' are

16.

YO

negative.

quadrant YOX' and P^M^, drawn
y, meet OX' in M^^ and if the

numerical values of the quantities OM^ and J/aPg be a
and h, the coordinates of P are {-a and h) and the position
of Pg is given by the symbol (—a, h).
If

P2 b® i^

*li®

parallel to the axis of

Similarly, if P3 be in the third quadrant X'OY', both of
coordinates are negative, and, if the numerical lengths
of Oi/3 and J/3P3 be c and d, then P3 is denoted by the

its

symbol (—

c,

Finally,

positive
17.

and
Ex.
(i)

—

d).

in the fourth quadrant its abscissa is
its ordinate is negative.

if

P4

lie

Lay down on "paper
(2,

-1),

(ii)

(-3,

the position of the points
2),

and

(iii)

(-2, -3).

To get the first point we measure a distance 2 along OX and then
a distance 1 parallel to OF'; we thus arrive at the required point.
To get the second point, we measure a distance 3 along OX', and
then 2 parallel to OY.
To get the third point, we measure 2 along OX' and then
3 parallel to OT.
These three points are respectively the points P4 P., and Pg in
,

,

the figure of Art. 15.

18. When the axes of coordinates are as in the figure
of Art. 15, not at right angles, they are said to be Oblique
Axes, and the angle between their two positive directions
and 07, i.e. the angle XOY, is generally denoted by

OX

the Greek letter

w.

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