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aontion: C. J. CLAY and SONS,

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,

AVE MARIA LANE.

(ffilagfloia: 263, ARGTLE STREET.

THE ELEMENTS

OP

COOEDINATE GEOMETRY.

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 CO.

AND NEW YOEK.

1895

[All Bights reserved.']

CTambrtlJse:

PRINTED BY J. & C. F. CLAY,

AT THE UNIVEBSITY PRESS.

150553

PKEFACE.

"TN the following work I have tried to present the

elements of Coordinate Geometry in a manner

suitable for Beginners and Junior Students. The

present book only deals with Cartesian and Polar

Coordinates. Within these limits I venture to hope

that the book is fairly complete, and that no proposi-

tions of very great importance have been omitted.

The Straight Line and Circle have been treated

more fully than the other portions of the subject,

since it is generally in the elementary conceptions

that beginners find great difficulties.

There are a large number of Examples, over 1100

in all, and they are, in general, of an elementary

character. The examples are especially numerous in

the earlier parts of the book.

vi PREFACE.

I am much indebted to several friends for reading

portions of the proof sheets, but especially to Mr W.

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

book and made many valuable suggestions.

For any criticisms, suggestions, or corrections, I

shall be grateful.

S. L. LONEY.

EoTAIi HOLLOWAY COLLEGE,

Egham, Surbey.

July 4, 1895.

CONTENTS.

CHAP. PAGE

I. Introduction. Algebraic Kesults ... 1

II. Coordinates. Lengths of Straight Lines and

Areas of Triangles 8

Polar Coordinates 19

III. Locus. Equation to a Locus 24

IV. The Straight Line. Eect angular Coordinates . 31

Straight line through two points .... 39

Angle between two given straight lines . . 42

Conditions that they may be parallel and per-

pendicular . . . . . . .44

Length of a perpendicular . . " . . 51

Bisectors of angles 58

V The Straight Line. Polar Equations and

Oblique Coordinates . . . . 66

Equations involving an arbitrary constant . . 73

Examples of loci 80

VI. Equations representing two or more Straight

Lines 88

Angle between two lines given by one equation 90

Greneral equation of the second degree . . 94

VII. Transformation of Coordinates . . . 109

Invariants 115

Vlii CONTENTS.

CHAP. PAGE

VIII. The Circle 118

Equation to a tangent 126

Pole and polar 137

Equation to a circle in polar coordinates . .145

Equation referred to oblique axes . . . 148

Equations in terms of one variable . . .150

IX. Systems of Circles 160

Orthogonal circles . . , . . . .160

Kadical axis 161

Coaxal circles 166

X. Conic Sections. The Parabola . 174

Equation to a tangent 180

Some properties of the parabola . . . 187

Pole and polar 190

Diameters 195

Equations in terms of one variable . . .198

XI. The Parabola {continued') .... 206

Loci connected with the parabola . . . 206

Three normals passing through a given point . 211

Parabola referred to two tangents as axes . .217

XII. The Ellipse 225

Auxiliary circle and eccentric angle . . .231

Equation to a tangent . . . . . 237

Some properties of the ellipse .... 242

Pole and polar 249

Conjugate diameters ...... 254

Pour normals through any point . . . 265

Examples of loci 266

XIII. The Hyperbola 271

Asymptotes 284

Equation referred to the asymptotes as axes . 296

One variable. Examples 299

CONTENTS. IX

CHAP. PAGE

XIV. Polar Equation to, a Conic .... 306

Polar equation to a tangent, polar, and normal , 313

XV. General Equation. Tracing of Curves . 322

Particular cases of conic sections .... 322

Transformation of equation to centre as origin 326

Equation to asymptotes 329

Tracing a parabola ...... 332

Tracing a central conic . . . . . . 338

Eccentricity and foci of general conic . 342

XVI. General Equation ...... 349

Tangent 349

Conjugate diameters ...... 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 ...... 364

The director circle ....... 365

The foci 367

The axes 369

Lengths of straight lines drawn in given directions

to meet the conic 370

Conies passing through four 23oints . . . 378

Conies touching four lines 380

â– The conic LM=B? 382

XVII. Miscellaneous Propositions .... 385

On the four normals from any point to a central

conic 385

Confocal conies ....... 392

Circles of curvature and contact of the third order . 398

Envelopes 407

Answers . i â€” xiii

ERKATA.

Page 87, Ex. 27, line 4. For "JR" read " S."

â€ž 235, Ex. 18, line 3. For "odd" read "even."

,, â€ž ,, ,, line 5. Dele "and Page 37, Ex. 15."

,, 282, Ex. 3. For "transverse" read "conjugate."

CHAPTER I.

INTRODUCTION.

SOME ALGEBRAIC RESULTS.

1. Quadratic Equations. The roots of the quad-

ratic equation

a'3^ + 6x + c =

may easily be shewn to be

- & + â€¢JlP' â€” 4ac 1 -b- s/b^ â€” 4:aG

2i. '^"'^ 2^ â€¢

They are therefore real and unequal, equal, or imaginary,

according as the quantity b^â€”iac is positive, zero, or negative,

i.e. according as b^ = 4:ac.

2. Relations between the roots of any algebraic equation

and the coejicients of the terms of the equation.

If any equation be written so that the coefficient of the

highest term is unity, it is shewn in any treatise on Algebra

that

(1) the sum of the roots is equal to the coefficient of

the second term with its sign changed,

(2) the sum of the products of the roots, taken two

at a time, is equal to the coefficient of the third term,

(3) the sum of their products, taken three at a time,

is equal to the coefficient of the fourth term with its sign

changed,

and so on.

L. e 1

COORDINATE GEOMETRY.

Ex. 1. If a and /3 be the roots of the equation

b c

ax'^ + bx + c = 0, i.e. x^ + - x + ~ = 0,

a a

we have

b -. ^ c

a + p= â€” and a^ = -

Ex. 2. If a, j8, and 7 be the roots of the cubic equation

ax^ + bx^ + cx + d=0,

i.e. of

we have

x^+-x^ +-x + - = 0,

a a a

a + p + y:

and

^y + ya + a^=:- ,

o-Pl-

3. It can easily be shewn that the solution of the

equations

a^x + h^y + G^z = 0,

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

IS

X

y

^1^2 ~ ^2^1 ^1^2 ~ ^2^1 '^1^2 ~ ^2^1

Determinant Notation.

4. The quantity-

is called a determinant of the

second order and stands for the quantity a-})^ â€” aj)^, so that

d-yf d^

^1, h

= Ob^^ â€” 6Â»2&i .

\%^\

Exs. (1) ;' | = 2x5-4x3 = 10-12=-2;

!4, 5i

3, -4|

(ii)

-7, -6

= - 3 X ( - 6) - { - 7) X ( - 4) = 18 - 28 = - 10.

DETERMINANTS.

5. The quantity

Â«!,

Â»2J

Â«3

^1,

&2J

^^3

Cl,

^2 5

^3

(1)

is called a determinant of the third order and stands for the

quantity

a. X

^2 J ^3

â€” a.

^2 5 <^3

&1, &.

+ Â«o

a> *^3i

61,62

(2),

i.e, by Art. 4, for the quantity

Â«i (^2^3 - ^3^2) - Â«^2 (^1^3 - &3C1) + ^3 (^i^^a - ^2^1)*

i.e. % (62C3 â€” h..G^ + (^2 (63C1 â€” 61C3) + Â«3 (61C2 â€” 62C1).

6. A determinant of the third order is therefore reduced

to three determinants of the second order by the following

rule :

Take in order the quantities which occur in the first row

of the determinant ; multiply each of these in turn by the

determinant which is obtained by erasing the row and

column to which it belongs ; prefix the sign + and â€” al-

ternately to the products thus obtained and add the

results.

Thus, if in (1) we omit the row and column to which a^

belongs, we have left the determinant ^'

^ i and this is the

coefficient of a-^ in (2).

Similarly, if in (1) we omit the row and column to which

a^ belongs, we have left the determinant ^'

and this

-'D

with the â€” sign prefixed is the coefficient of a^ in (2).

7. Ex.

The determinant

1,

-4,

-7,

-2, -3

5,-6

8, -9

X

5,-6

8,-9

-(-2)x

-4,

-7,

- W

â– 3)x

-4,5

-7,8

= {5x(-9)-8x(-6)}+2x{(-4)(-9)-(-7)(-6)}

-3x{(-4)x8-(-7)x5}

= {-45 + 48} +2(36-42} -3 {-32 + 35}

= 3-12-9= -18.

1â€”2

COORDINATE GEOMETRY.

8. The quantity

(h.1 ^2> %J ^4

61, &2) hi h

^11 ^25 ^3>

j ^1) ^2 5 ^3) ^4

is called a determinant of the fourth order and stands for

the quantity

Â«i X

K h, ^4

^2Â» ^3 J

<^2> ^3> ^4

i^lJ ^35 h

â€” Clo X \ C-,

^3}

+ 6^3 X

1 1 5 3 3 4

&i, 62J ^4!

C^ cCj_ X

1 ? 2 5 4

&1,

<^2, h

Cl,

^2) Cg

c?i,

Â»2J <^3

and its value may be obtained by finding the value of each

of these four determinants by the rule of Art. 6.

The rule for finding the value of a determinant of the

fourth order in terms of determinants of the third order is

clearly the same as that for one of the third order given in

Art. 6.

Similarly for determinants of higher orders.

9. A determinant of the second order has two terms.

One of the third order has 3x2, i.e. 6, terms. One of the

fourth order has 4 x 3 x 2, -i.e. 24, terms, and so on.

(1)

(4)

(6)

10. Exs.

2, -3

4, 8

Prove that

= 28. (2)

9, 8, 7j

6, 5, 4 =0.

3, 2, l|

a, h, g

-6,

-4.

7

-9

= 85

!.. (3)

5,

-2,

9,

-3, 7

4,-8

3, -10

-a, b, c

(5)

a, -b, c

=:4a6c.

a, I

, -c

-98.

9, f, c

= abc + 2fgh - ap - bg^ - ch\

ELIMINATION. 5

Elimination.

11. Suppose we have the two equations

aj^x + a^y = (1),

\x +b^y ^0 (2),

between the two unknown quantities x and y. There must

be some relation holding between the four coefficients 6*i, ctaj

bi, and 63 â€¢ ^or, from (1), we have

y~ %'

and, from (2), we have - = â€” =-^ .

y K

X

Equating these two values of - we have

i.e. a-J)^ â€” ajb^ = (3).

The result (3) is the condition that both the equations

(1) and (2) should be true for the same values of x and y.

The process of finding this condition is called the elimi-

nating 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 ) '^

This result is obtained from (1) and (2) by taking the

coefficients of x and y in the order in which they occur in

the equations, placing them in this order to form a determi-

nant, and equating it to zero.

written in the form

0.

12. Suppose, again, that we have the three equations

a-^x + a^y + a^^ = (1),

\x+ h^y^ h^z = (2),

and G^x + G^y + C3S = (3),

between the three unknown quantities x, y, and z.

6

COORDINATE GEOMETRY.

By dividing each equation by z we have three equations

X

y

between the two unknown quantities â€” and -

z z

Two of

%,

^2,

%

&1,

\y

h

Ci,

^2 1

Cs

these will be sufficient to determine these quantities. By

substituting their values in the third equation we shall

obtain a relation between the nine coefficients.

Or we may proceed thus. From the equations (2) and

(3) we have

X __ y _ ^

Substituting these values in (1), we have

Â«1 (^2^3 - ^3^2) + Â«2 (^3^1 - ^1^3) + Â«3 (^1^2 - ^2^1) = 0. . .(4).

This is the result of eliminating cc, 3/, and % from the

equations (1), (2), and (3).

But, by Art. 5, equation (4) may be written in the form

= 0.

This eliminant may be written down as in the last

article, viz. by taking the coefficients of x, y, and z in the

order in which they occur in the equations (1), (2), and (3),

placing them to form a determinant, and equating it to

zero.

13. Ex. What is the value of a so that the equations

ax + 2y + 3z = 0, 2x-3y + 4:Z = 0,

and 5x + 7y-8z=:0

may be simultaneously true ?

Eliminating x, y, and z, we have

a, 2, 3,

2, -3, 41 = 0,

5, 7, -8!

^.e. Â« [( - 3) ( - 8) - 4 X 7] - 2 [2 X { - 8) - 4 X 5] + 3 [2 X 7 - 5 X ( - 3)]=0,

i.e. Â«[-4]-2[-36] + 3[29] = 0,

^, ^ 72 + 87 159

so that a= â€” -, = â€” ;- .

4 4

ELIMINATION.

14. If again we have the four equations

a-^x + dil/ + cf'zZ + a^u = 0,

h^x + h^y + b^z + b^u = 0,

CiÂ«; + c^i/ + G^z + c^u = 0,

and djX + d^y + d.^z + d^ â€” 0,

it could be shewn that the result of eliminating the four

quantities cc, y, z^ and u is the determinant

Â«1J

^2,

%,

Â«4

\.

^2,

bz,

^>4

Ci,

^2?

^it

C4

c?i,

C?2,

C?3,

c?.

A similar theorem could be shewn to be true for n

equations of the first degree, such as the above, between

n unknown quantities.

It will be noted that the right-hand member of each of

the above equations is zero.

CHAPTER II.

COORDINATES. LENGTHS OF STRAIGHT LINES AND

AREAS OF TRIANGLES.

15. Coordinates. Let OX and 07 be two fixed

straight lines in the plane of the paper. The line OX is

called the axis of cc, the line OY the axis of y, whilst the

two together are called the axes of coordinates.

The point is called the origin of coordinates or, more

shortly, the origin.

From any point F in the

plane draw a straight line

parallel to OF to meet OX

in M.

The distance OM is called

the Abscissa, and the distance

MP the Ordinate of the point

P, whilst the abscissa and the

ordinate together are called

its Coordinates.

Distances measured parallel to OX are called a?, with

or without a suffix, {e.g.Xj, x.-^... x\ x",...), and distances

measured parallel to OY are called y, with or without a

suffix, (e.g. 2/i, 2/2, - - 2/'. y", - -)-

If the distances OM and MP be respectively x and ?/,

the coordinates of P are, for brevity, denoted by the symbol

{x, y).

Conversely, when we are given that the coordinates of

a point P are (x, y) we know its position. For from we

have only to measure a distance OM {â€”x) along OX and

COORDINATES. 9

then from 21 measure a distance MP {=y) parallel to OY

and we arrive at the position of the point P. For example

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

MP= WM, then P is the point (1, 2).

16. Produce XO backwards to form the line OX' and

YO 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

negative.

If P2 bÂ® i^ *liÂ® quadrant YOX' and P^M^, drawn

parallel to the axis of 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 (â€”a, h).

Similarly, if P3 be in the third quadrant X'OY', both of

its coordinates are negative, and, if the numerical lengths

of Oi/3 and J/3P3 be c and d, then P3 is denoted by the

symbol (â€” c, â€” d).

Finally, if P4 lie in the fourth quadrant its abscissa is

positive and its ordinate is negative.

17. Ex. Lay down on "paper the position of the points

(i) (2, -1), (ii) (-3, 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

OX and 07, i.e. the angle XOY, is generally denoted by

the Greek letter w.

10 COORDINATE GEOMETRY.

In general, it is however found to be more convenient to

take the axes OX and OZat right angles. They are then

said to be Rectangular Axes.

It may always be assumed throughout this book that

the axes are rectangular unless it is otherwise stated.

19. The system of coordinates spoken of in the last

few articles is known as the Cartesian System of Coordi-

nates. It is so called because this system was first intro-

duced by the philosopher Des Cartes. There are other

systems of coordinates in use, but the Cartesian system is

by far the most important.

20. To find the distance between two points whose co-

ordinates are given.

Let Pi and P^ be the two

given points, and let their co-

ordinates be respectively {x^ , y^)

and (a^sj 2/2)-

Draw Pji/i and P^M^ pa-

rallel to OY, to meet OX in

J/j and M^. Draw P^R parallel

to OX to meet M-^P^ in R. q ' M jvT

Then

P^R = M^Mt^ = OM^ - OMc^ = oi^-X2,

RP, = M,P,-M,P, = y,~y,,

and z P^i^Pi = z6>ifiPa-l 80Â° -PiJfiX^l 80Â° -<o.

We therefore have [Trigonometry, Art. 164]

P^P^^ = P^R^ + RP^^ - 2P^R . PPi cos P^RP^

- (^1 - x^Y + (2/1 - 2/2)' - 2 (a^i - x^) (2/1 - 2/2) cos (180Â° - (o)

= (Xi-X2)2 + (yj_y2)2+2(Xi-X2)(yi-y2)COSCO...(l).

If the axes be, as is generally the case, at right angles,

we have <o == 90Â° and hence cos to = 0.

The formula (1) then becomes

P^P^ - (x, - x^Y + (2/1 - y^Y^

DISTANCE BETWEEN TWO POINTS.

11

SO that in rectangular coordinates the distance between the

two points (x^j y^ and (a-g, 2/2) is

V(Xi - x^)^ + (Yi - y^)^ (2).

Cor. The distance of the point (x^, y-^ from the origin

is Jx^ + 2/1^, the axes being rectangular. This follows from

(2) by making both x^ and y^ equal to zero.

21. The formula of the previous article has been proved for the

case when the coordinates of both the points are all positive.!

Due regard being had to the signs of the coordinates, the formula

will be found to be true for all

points.

As a numerical example, let

Pj be the point (5, 6) and Pg

be the point (-7, -4), so that

we have

and y2= -^.

Then

P^ = 31^0 + OM^ = 7 + 5

and

RPt^ = EM-^ + l/iPj = 4 + 6

= -2/2 + 2/1.

The rest of the proof is as in the last article.

Similarly any other case could be considered.

22. To find tJie coordinates of the point which divides

in a given ratio (ni^ : m^ the line joining two given jyoints

(a?!, 2/1) and (x^, y^).

Yi

O M, M

M,

X

Let Pi be the point {x^, y^), Po the point (x^, y^), and P

the required point, so that we have

12 COORDINATE GEOMETRY.

Let P be the point (sc, y) so that if P^M^, PM, and

P^M^ be drawn parallel to the axis of y to meet the axis of

Â£C in i/i, Mj and M^, we have

Oi/i = Â£Ci, M^P^ = y^, OM=x, MP = y, QM^^x^,

and i/^z^a = 2/2-

Draw PiEi and P-Sg, parallel to OX, to meet J/P and

M^P^ in Pi and Pg respectively.

Then PjPi = M^^M^^ OM- OM^ = x-x^,

PR^ = MM^ = OJ/2 - 0M= x,^ - X,

R,P^MP-M,P, = y-y,,

and P2P2 = M^P^ - MP = y^-y.

. From the similar triangles PiPjP and PR^P^ we have

m^ PjP PiRi X â€” Xt^

m^ PP^ PR^ x^ â€” x'

, ifv-tt/Uey *T" i/VoOO-i

t.e. x = ^^ .

Again

mi P,P R,P y-y.

- * m^ PP2 P2P2 2/2-2/'

so that mi (3/2 - 3/) = 7^2 {y - 3/1),

and hence y = -^^ ^-^ .

Wi + 7?22

The coordinates of the point which divides PiP^ in-

ternally in the given ratio rrii : tyi^ are therefore

nil + ^2 mi + nig *

If the point Q divide the line P1P2 externally in the

same ratio, i.e. so that P^Q : QP^ :: mj : m^i its coordinates

would be found to be

nil "" ^^2 '^i "â– ^^2

The proof of this statement is similar to that of the

preceding article and is left as an exercise for the student.

LINES DIVIDED IN A GIVEN RATIO. 13

Cor. The coordinates of the middle point of the line

joining {x^, y^ to {x^, y^ are

23. Ex. 1. In any triangle ABC 'prove that

AB^ + AC^ = 2 {AD^ + DG^),

lohere D is the middle point of BG.

Take B as origin, 5C as the axis of x, and a line through B i>er-

pendicular to BC as the axis of y.

Let BG=a, so that G is the point (a, 0), and let A be the point

Then D is the point (|> C> j .

Hence ^D2=ra;i -^Y + i/i^ and DG^=f~y.

Hence 2 (^D^ + DC^) ::= 2 ["x^^ + y^^ - ax^ + ^~|

= 2xi2 + 2yi2_2o.x.^ + a2.

Also ^C'2.= (a;i-a)2 + ?j^2^

and AB^=^x^-\-y^.

Therefore AB'^ + ^(72 = 'Ix^ + 2?/i2 _ 2aa;i + a^.

Hence ^52 + ^(72^2(^2)2 + 2)(72)_

This is the well-known theorem of Ptolemy.

Ex. 2. ABG is a triangle and D, E, and F are the middle points

of the sides BG, GA, and AB ; prove that the point lohich divides AD

internally in the ratio 2 : 1 also divides the lines BE and GF in

uTd-nu

/o-

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aontion: C. J. CLAY and SONS,

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,

AVE MARIA LANE.

(ffilagfloia: 263, ARGTLE STREET.

THE ELEMENTS

OP

COOEDINATE GEOMETRY.

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 CO.

AND NEW YOEK.

1895

[All Bights reserved.']

CTambrtlJse:

PRINTED BY J. & C. F. CLAY,

AT THE UNIVEBSITY PRESS.

150553

PKEFACE.

"TN the following work I have tried to present the

elements of Coordinate Geometry in a manner

suitable for Beginners and Junior Students. The

present book only deals with Cartesian and Polar

Coordinates. Within these limits I venture to hope

that the book is fairly complete, and that no proposi-

tions of very great importance have been omitted.

The Straight Line and Circle have been treated

more fully than the other portions of the subject,

since it is generally in the elementary conceptions

that beginners find great difficulties.

There are a large number of Examples, over 1100

in all, and they are, in general, of an elementary

character. The examples are especially numerous in

the earlier parts of the book.

vi PREFACE.

I am much indebted to several friends for reading

portions of the proof sheets, but especially to Mr W.

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

book and made many valuable suggestions.

For any criticisms, suggestions, or corrections, I

shall be grateful.

S. L. LONEY.

EoTAIi HOLLOWAY COLLEGE,

Egham, Surbey.

July 4, 1895.

CONTENTS.

CHAP. PAGE

I. Introduction. Algebraic Kesults ... 1

II. Coordinates. Lengths of Straight Lines and

Areas of Triangles 8

Polar Coordinates 19

III. Locus. Equation to a Locus 24

IV. The Straight Line. Eect angular Coordinates . 31

Straight line through two points .... 39

Angle between two given straight lines . . 42

Conditions that they may be parallel and per-

pendicular . . . . . . .44

Length of a perpendicular . . " . . 51

Bisectors of angles 58

V The Straight Line. Polar Equations and

Oblique Coordinates . . . . 66

Equations involving an arbitrary constant . . 73

Examples of loci 80

VI. Equations representing two or more Straight

Lines 88

Angle between two lines given by one equation 90

Greneral equation of the second degree . . 94

VII. Transformation of Coordinates . . . 109

Invariants 115

Vlii CONTENTS.

CHAP. PAGE

VIII. The Circle 118

Equation to a tangent 126

Pole and polar 137

Equation to a circle in polar coordinates . .145

Equation referred to oblique axes . . . 148

Equations in terms of one variable . . .150

IX. Systems of Circles 160

Orthogonal circles . . , . . . .160

Kadical axis 161

Coaxal circles 166

X. Conic Sections. The Parabola . 174

Equation to a tangent 180

Some properties of the parabola . . . 187

Pole and polar 190

Diameters 195

Equations in terms of one variable . . .198

XI. The Parabola {continued') .... 206

Loci connected with the parabola . . . 206

Three normals passing through a given point . 211

Parabola referred to two tangents as axes . .217

XII. The Ellipse 225

Auxiliary circle and eccentric angle . . .231

Equation to a tangent . . . . . 237

Some properties of the ellipse .... 242

Pole and polar 249

Conjugate diameters ...... 254

Pour normals through any point . . . 265

Examples of loci 266

XIII. The Hyperbola 271

Asymptotes 284

Equation referred to the asymptotes as axes . 296

One variable. Examples 299

CONTENTS. IX

CHAP. PAGE

XIV. Polar Equation to, a Conic .... 306

Polar equation to a tangent, polar, and normal , 313

XV. General Equation. Tracing of Curves . 322

Particular cases of conic sections .... 322

Transformation of equation to centre as origin 326

Equation to asymptotes 329

Tracing a parabola ...... 332

Tracing a central conic . . . . . . 338

Eccentricity and foci of general conic . 342

XVI. General Equation ...... 349

Tangent 349

Conjugate diameters ...... 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 ...... 364

The director circle ....... 365

The foci 367

The axes 369

Lengths of straight lines drawn in given directions

to meet the conic 370

Conies passing through four 23oints . . . 378

Conies touching four lines 380

â– The conic LM=B? 382

XVII. Miscellaneous Propositions .... 385

On the four normals from any point to a central

conic 385

Confocal conies ....... 392

Circles of curvature and contact of the third order . 398

Envelopes 407

Answers . i â€” xiii

ERKATA.

Page 87, Ex. 27, line 4. For "JR" read " S."

â€ž 235, Ex. 18, line 3. For "odd" read "even."

,, â€ž ,, ,, line 5. Dele "and Page 37, Ex. 15."

,, 282, Ex. 3. For "transverse" read "conjugate."

CHAPTER I.

INTRODUCTION.

SOME ALGEBRAIC RESULTS.

1. Quadratic Equations. The roots of the quad-

ratic equation

a'3^ + 6x + c =

may easily be shewn to be

- & + â€¢JlP' â€” 4ac 1 -b- s/b^ â€” 4:aG

2i. '^"'^ 2^ â€¢

They are therefore real and unequal, equal, or imaginary,

according as the quantity b^â€”iac is positive, zero, or negative,

i.e. according as b^ = 4:ac.

2. Relations between the roots of any algebraic equation

and the coejicients of the terms of the equation.

If any equation be written so that the coefficient of the

highest term is unity, it is shewn in any treatise on Algebra

that

(1) the sum of the roots is equal to the coefficient of

the second term with its sign changed,

(2) the sum of the products of the roots, taken two

at a time, is equal to the coefficient of the third term,

(3) the sum of their products, taken three at a time,

is equal to the coefficient of the fourth term with its sign

changed,

and so on.

L. e 1

COORDINATE GEOMETRY.

Ex. 1. If a and /3 be the roots of the equation

b c

ax'^ + bx + c = 0, i.e. x^ + - x + ~ = 0,

a a

we have

b -. ^ c

a + p= â€” and a^ = -

Ex. 2. If a, j8, and 7 be the roots of the cubic equation

ax^ + bx^ + cx + d=0,

i.e. of

we have

x^+-x^ +-x + - = 0,

a a a

a + p + y:

and

^y + ya + a^=:- ,

o-Pl-

3. It can easily be shewn that the solution of the

equations

a^x + h^y + G^z = 0,

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

IS

X

y

^1^2 ~ ^2^1 ^1^2 ~ ^2^1 '^1^2 ~ ^2^1

Determinant Notation.

4. The quantity-

is called a determinant of the

second order and stands for the quantity a-})^ â€” aj)^, so that

d-yf d^

^1, h

= Ob^^ â€” 6Â»2&i .

\%^\

Exs. (1) ;' | = 2x5-4x3 = 10-12=-2;

!4, 5i

3, -4|

(ii)

-7, -6

= - 3 X ( - 6) - { - 7) X ( - 4) = 18 - 28 = - 10.

DETERMINANTS.

5. The quantity

Â«!,

Â»2J

Â«3

^1,

&2J

^^3

Cl,

^2 5

^3

(1)

is called a determinant of the third order and stands for the

quantity

a. X

^2 J ^3

â€” a.

^2 5 <^3

&1, &.

+ Â«o

a> *^3i

61,62

(2),

i.e, by Art. 4, for the quantity

Â«i (^2^3 - ^3^2) - Â«^2 (^1^3 - &3C1) + ^3 (^i^^a - ^2^1)*

i.e. % (62C3 â€” h..G^ + (^2 (63C1 â€” 61C3) + Â«3 (61C2 â€” 62C1).

6. A determinant of the third order is therefore reduced

to three determinants of the second order by the following

rule :

Take in order the quantities which occur in the first row

of the determinant ; multiply each of these in turn by the

determinant which is obtained by erasing the row and

column to which it belongs ; prefix the sign + and â€” al-

ternately to the products thus obtained and add the

results.

Thus, if in (1) we omit the row and column to which a^

belongs, we have left the determinant ^'

^ i and this is the

coefficient of a-^ in (2).

Similarly, if in (1) we omit the row and column to which

a^ belongs, we have left the determinant ^'

and this

-'D

with the â€” sign prefixed is the coefficient of a^ in (2).

7. Ex.

The determinant

1,

-4,

-7,

-2, -3

5,-6

8, -9

X

5,-6

8,-9

-(-2)x

-4,

-7,

- W

â– 3)x

-4,5

-7,8

= {5x(-9)-8x(-6)}+2x{(-4)(-9)-(-7)(-6)}

-3x{(-4)x8-(-7)x5}

= {-45 + 48} +2(36-42} -3 {-32 + 35}

= 3-12-9= -18.

1â€”2

COORDINATE GEOMETRY.

8. The quantity

(h.1 ^2> %J ^4

61, &2) hi h

^11 ^25 ^3>

j ^1) ^2 5 ^3) ^4

is called a determinant of the fourth order and stands for

the quantity

Â«i X

K h, ^4

^2Â» ^3 J

<^2> ^3> ^4

i^lJ ^35 h

â€” Clo X \ C-,

^3}

+ 6^3 X

1 1 5 3 3 4

&i, 62J ^4!

C^ cCj_ X

1 ? 2 5 4

&1,

<^2, h

Cl,

^2) Cg

c?i,

Â»2J <^3

and its value may be obtained by finding the value of each

of these four determinants by the rule of Art. 6.

The rule for finding the value of a determinant of the

fourth order in terms of determinants of the third order is

clearly the same as that for one of the third order given in

Art. 6.

Similarly for determinants of higher orders.

9. A determinant of the second order has two terms.

One of the third order has 3x2, i.e. 6, terms. One of the

fourth order has 4 x 3 x 2, -i.e. 24, terms, and so on.

(1)

(4)

(6)

10. Exs.

2, -3

4, 8

Prove that

= 28. (2)

9, 8, 7j

6, 5, 4 =0.

3, 2, l|

a, h, g

-6,

-4.

7

-9

= 85

!.. (3)

5,

-2,

9,

-3, 7

4,-8

3, -10

-a, b, c

(5)

a, -b, c

=:4a6c.

a, I

, -c

-98.

9, f, c

= abc + 2fgh - ap - bg^ - ch\

ELIMINATION. 5

Elimination.

11. Suppose we have the two equations

aj^x + a^y = (1),

\x +b^y ^0 (2),

between the two unknown quantities x and y. There must

be some relation holding between the four coefficients 6*i, ctaj

bi, and 63 â€¢ ^or, from (1), we have

y~ %'

and, from (2), we have - = â€” =-^ .

y K

X

Equating these two values of - we have

i.e. a-J)^ â€” ajb^ = (3).

The result (3) is the condition that both the equations

(1) and (2) should be true for the same values of x and y.

The process of finding this condition is called the elimi-

nating 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 ) '^

This result is obtained from (1) and (2) by taking the

coefficients of x and y in the order in which they occur in

the equations, placing them in this order to form a determi-

nant, and equating it to zero.

written in the form

0.

12. Suppose, again, that we have the three equations

a-^x + a^y + a^^ = (1),

\x+ h^y^ h^z = (2),

and G^x + G^y + C3S = (3),

between the three unknown quantities x, y, and z.

6

COORDINATE GEOMETRY.

By dividing each equation by z we have three equations

X

y

between the two unknown quantities â€” and -

z z

Two of

%,

^2,

%

&1,

\y

h

Ci,

^2 1

Cs

these will be sufficient to determine these quantities. By

substituting their values in the third equation we shall

obtain a relation between the nine coefficients.

Or we may proceed thus. From the equations (2) and

(3) we have

X __ y _ ^

Substituting these values in (1), we have

Â«1 (^2^3 - ^3^2) + Â«2 (^3^1 - ^1^3) + Â«3 (^1^2 - ^2^1) = 0. . .(4).

This is the result of eliminating cc, 3/, and % from the

equations (1), (2), and (3).

But, by Art. 5, equation (4) may be written in the form

= 0.

This eliminant may be written down as in the last

article, viz. by taking the coefficients of x, y, and z in the

order in which they occur in the equations (1), (2), and (3),

placing them to form a determinant, and equating it to

zero.

13. Ex. What is the value of a so that the equations

ax + 2y + 3z = 0, 2x-3y + 4:Z = 0,

and 5x + 7y-8z=:0

may be simultaneously true ?

Eliminating x, y, and z, we have

a, 2, 3,

2, -3, 41 = 0,

5, 7, -8!

^.e. Â« [( - 3) ( - 8) - 4 X 7] - 2 [2 X { - 8) - 4 X 5] + 3 [2 X 7 - 5 X ( - 3)]=0,

i.e. Â«[-4]-2[-36] + 3[29] = 0,

^, ^ 72 + 87 159

so that a= â€” -, = â€” ;- .

4 4

ELIMINATION.

14. If again we have the four equations

a-^x + dil/ + cf'zZ + a^u = 0,

h^x + h^y + b^z + b^u = 0,

CiÂ«; + c^i/ + G^z + c^u = 0,

and djX + d^y + d.^z + d^ â€” 0,

it could be shewn that the result of eliminating the four

quantities cc, y, z^ and u is the determinant

Â«1J

^2,

%,

Â«4

\.

^2,

bz,

^>4

Ci,

^2?

^it

C4

c?i,

C?2,

C?3,

c?.

A similar theorem could be shewn to be true for n

equations of the first degree, such as the above, between

n unknown quantities.

It will be noted that the right-hand member of each of

the above equations is zero.

CHAPTER II.

COORDINATES. LENGTHS OF STRAIGHT LINES AND

AREAS OF TRIANGLES.

15. Coordinates. Let OX and 07 be two fixed

straight lines in the plane of the paper. The line OX is

called the axis of cc, the line OY the axis of y, whilst the

two together are called the axes of coordinates.

The point is called the origin of coordinates or, more

shortly, the origin.

From any point F in the

plane draw a straight line

parallel to OF to meet OX

in M.

The distance OM is called

the Abscissa, and the distance

MP the Ordinate of the point

P, whilst the abscissa and the

ordinate together are called

its Coordinates.

Distances measured parallel to OX are called a?, with

or without a suffix, {e.g.Xj, x.-^... x\ x",...), and distances

measured parallel to OY are called y, with or without a

suffix, (e.g. 2/i, 2/2, - - 2/'. y", - -)-

If the distances OM and MP be respectively x and ?/,

the coordinates of P are, for brevity, denoted by the symbol

{x, y).

Conversely, when we are given that the coordinates of

a point P are (x, y) we know its position. For from we

have only to measure a distance OM {â€”x) along OX and

COORDINATES. 9

then from 21 measure a distance MP {=y) parallel to OY

and we arrive at the position of the point P. For example

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

MP= WM, then P is the point (1, 2).

16. Produce XO backwards to form the line OX' and

YO 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

negative.

If P2 bÂ® i^ *liÂ® quadrant YOX' and P^M^, drawn

parallel to the axis of 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 (â€”a, h).

Similarly, if P3 be in the third quadrant X'OY', both of

its coordinates are negative, and, if the numerical lengths

of Oi/3 and J/3P3 be c and d, then P3 is denoted by the

symbol (â€” c, â€” d).

Finally, if P4 lie in the fourth quadrant its abscissa is

positive and its ordinate is negative.

17. Ex. Lay down on "paper the position of the points

(i) (2, -1), (ii) (-3, 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

OX and 07, i.e. the angle XOY, is generally denoted by

the Greek letter w.

10 COORDINATE GEOMETRY.

In general, it is however found to be more convenient to

take the axes OX and OZat right angles. They are then

said to be Rectangular Axes.

It may always be assumed throughout this book that

the axes are rectangular unless it is otherwise stated.

19. The system of coordinates spoken of in the last

few articles is known as the Cartesian System of Coordi-

nates. It is so called because this system was first intro-

duced by the philosopher Des Cartes. There are other

systems of coordinates in use, but the Cartesian system is

by far the most important.

20. To find the distance between two points whose co-

ordinates are given.

Let Pi and P^ be the two

given points, and let their co-

ordinates be respectively {x^ , y^)

and (a^sj 2/2)-

Draw Pji/i and P^M^ pa-

rallel to OY, to meet OX in

J/j and M^. Draw P^R parallel

to OX to meet M-^P^ in R. q ' M jvT

Then

P^R = M^Mt^ = OM^ - OMc^ = oi^-X2,

RP, = M,P,-M,P, = y,~y,,

and z P^i^Pi = z6>ifiPa-l 80Â° -PiJfiX^l 80Â° -<o.

We therefore have [Trigonometry, Art. 164]

P^P^^ = P^R^ + RP^^ - 2P^R . PPi cos P^RP^

- (^1 - x^Y + (2/1 - 2/2)' - 2 (a^i - x^) (2/1 - 2/2) cos (180Â° - (o)

= (Xi-X2)2 + (yj_y2)2+2(Xi-X2)(yi-y2)COSCO...(l).

If the axes be, as is generally the case, at right angles,

we have <o == 90Â° and hence cos to = 0.

The formula (1) then becomes

P^P^ - (x, - x^Y + (2/1 - y^Y^

DISTANCE BETWEEN TWO POINTS.

11

SO that in rectangular coordinates the distance between the

two points (x^j y^ and (a-g, 2/2) is

V(Xi - x^)^ + (Yi - y^)^ (2).

Cor. The distance of the point (x^, y-^ from the origin

is Jx^ + 2/1^, the axes being rectangular. This follows from

(2) by making both x^ and y^ equal to zero.

21. The formula of the previous article has been proved for the

case when the coordinates of both the points are all positive.!

Due regard being had to the signs of the coordinates, the formula

will be found to be true for all

points.

As a numerical example, let

Pj be the point (5, 6) and Pg

be the point (-7, -4), so that

we have

and y2= -^.

Then

P^ = 31^0 + OM^ = 7 + 5

and

RPt^ = EM-^ + l/iPj = 4 + 6

= -2/2 + 2/1.

The rest of the proof is as in the last article.

Similarly any other case could be considered.

22. To find tJie coordinates of the point which divides

in a given ratio (ni^ : m^ the line joining two given jyoints

(a?!, 2/1) and (x^, y^).

Yi

O M, M

M,

X

Let Pi be the point {x^, y^), Po the point (x^, y^), and P

the required point, so that we have

12 COORDINATE GEOMETRY.

Let P be the point (sc, y) so that if P^M^, PM, and

P^M^ be drawn parallel to the axis of y to meet the axis of

Â£C in i/i, Mj and M^, we have

Oi/i = Â£Ci, M^P^ = y^, OM=x, MP = y, QM^^x^,

and i/^z^a = 2/2-

Draw PiEi and P-Sg, parallel to OX, to meet J/P and

M^P^ in Pi and Pg respectively.

Then PjPi = M^^M^^ OM- OM^ = x-x^,

PR^ = MM^ = OJ/2 - 0M= x,^ - X,

R,P^MP-M,P, = y-y,,

and P2P2 = M^P^ - MP = y^-y.

. From the similar triangles PiPjP and PR^P^ we have

m^ PjP PiRi X â€” Xt^

m^ PP^ PR^ x^ â€” x'

, ifv-tt/Uey *T" i/VoOO-i

t.e. x = ^^ .

Again

mi P,P R,P y-y.

- * m^ PP2 P2P2 2/2-2/'

so that mi (3/2 - 3/) = 7^2 {y - 3/1),

and hence y = -^^ ^-^ .

Wi + 7?22

The coordinates of the point which divides PiP^ in-

ternally in the given ratio rrii : tyi^ are therefore

nil + ^2 mi + nig *

If the point Q divide the line P1P2 externally in the

same ratio, i.e. so that P^Q : QP^ :: mj : m^i its coordinates

would be found to be

nil "" ^^2 '^i "â– ^^2

The proof of this statement is similar to that of the

preceding article and is left as an exercise for the student.

LINES DIVIDED IN A GIVEN RATIO. 13

Cor. The coordinates of the middle point of the line

joining {x^, y^ to {x^, y^ are

23. Ex. 1. In any triangle ABC 'prove that

AB^ + AC^ = 2 {AD^ + DG^),

lohere D is the middle point of BG.

Take B as origin, 5C as the axis of x, and a line through B i>er-

pendicular to BC as the axis of y.

Let BG=a, so that G is the point (a, 0), and let A be the point

Then D is the point (|> C> j .

Hence ^D2=ra;i -^Y + i/i^ and DG^=f~y.

Hence 2 (^D^ + DC^) ::= 2 ["x^^ + y^^ - ax^ + ^~|

= 2xi2 + 2yi2_2o.x.^ + a2.

Also ^C'2.= (a;i-a)2 + ?j^2^

and AB^=^x^-\-y^.

Therefore AB'^ + ^(72 = 'Ix^ + 2?/i2 _ 2aa;i + a^.

Hence ^52 + ^(72^2(^2)2 + 2)(72)_

This is the well-known theorem of Ptolemy.

Ex. 2. ABG is a triangle and D, E, and F are the middle points

of the sides BG, GA, and AB ; prove that the point lohich divides AD

internally in the ratio 2 : 1 also divides the lines BE and GF in

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