S. L. (Sidney Luxton) Loney.

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aontion: C. J. CLAY and SONS,



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S. L. LONEY, M.A.,


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"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.


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.



Egham, Surbey.
July 4, 1895.



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



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



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

■ 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 — xiii


Page 87, Ex. 27, line 4. For "JR" read " S."
„ 235, Ex. 18, line 3. For "odd" read "even."
,, „ ,, ,, line 5. Dele "and Page 37, Ex. 15."
,, 282, Ex. 3. For "transverse" read "conjugate."




1. Quadratic Equations. The roots of the quad-
ratic equation

a'3^ + 6x + c =

may easily be shewn to be

- & + •JlP' — 4ac 1 -b- s/b^ — 4:aG
2i. '^"'^ 2^ •

They are therefore real and unequal, equal, or imaginary,
according as the quantity b^—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

(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

and so on.

L. e 1


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= — 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:


^y + ya + a^=:- ,


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

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

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




^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-})^ — aj)^, so that

d-yf d^
^1, h

= Ob^^ — 6»2&i .


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

!4, 5i

3, -4|


-7, -6

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


5. The quantity








^2 5



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

a. X

^2 J ^3

— a.

^2 5 <^3

&1, &.

+ «o

a> *^3i



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

«i (^2^3 - ^3^2) - «^2 (^1^3 - &3C1) + ^3 (^i^^a - ^2^1)*
i.e. % (62C3 — h..G^ + (^2 (63C1 — 61C3) + «3 (61C2 — 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 — al-
ternately to the products thus obtained and add the

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


with the — sign prefixed is the coefficient of a^ in (2).

7. Ex.

The determinant



-2, -3

8, -9







- W




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

= {-45 + 48} +2(36-42} -3 {-32 + 35}
= 3-12-9= -18.



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

«i X

K h, ^4

^2» ^3 J

<^2> ^3> ^4

i^lJ ^35 h

— Clo X \ C-,


+ 6^3 X

1 1 5 3 3 4

&i, 62J ^4!

C^ cCj_ X

1 ? 2 5 4


<^2, h


^2) Cg


»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.




10. Exs.

2, -3

4, 8

Prove that
= 28. (2)

9, 8, 7j
6, 5, 4 =0.
3, 2, l|

a, h, g




= 85

!.. (3)




-3, 7
3, -10

-a, b, c


a, -b, c


a, I

, -c


9, f, c

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



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 • ^or, from (1), we have

y~ %'

and, from (2), we have - = — =-^ .

y K


Equating these two values of - we have

i.e. a-J)^ — 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


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.



By dividing each equation by z we have three equations



between the two unknown quantities — and -

z z

Two of








^2 1


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

«1 (^2^3 - ^3^2) + «2 (^3^1 - ^1^3) + «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

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. « [( - 3) ( - 8) - 4 X 7] - 2 [2 X { - 8) - 4 X 5] + 3 [2 X 7 - 5 X ( - 3)]=0,
i.e. «[-4]-2[-36] + 3[29] = 0,

^, ^ 72 + 87 159

so that a= — -, = — ;- .

4 4


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,

Ci«; + c^i/ + G^z + c^u = 0,

and djX + d^y + d.^z + d^ — 0,

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

















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.



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 {—x) along OX and


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

If P2 b® i^ *li® 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 (—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 (— c, — 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.


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


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

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

and z P^i^Pi = z6>ifiPa-l 80° -PiJfiX^l 80° -<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 (180° - (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 == 90° and hence cos to = 0.

The formula (1) then becomes

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



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

As a numerical example, let
Pj be the point (5, 6) and Pg
be the point (-7, -4), so that
we have

and y2= -^.

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


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^).


O M, M



Let Pi be the point {x^, y^), Po the point (x^, y^), and P
the required point, so that we have


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
£C in i/i, Mj and M^, we have

Oi/i = £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 — Xt^
m^ PP^ PR^ x^ — x'

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

t.e. x = ^^ .


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 "■ ^^2

The proof of this statement is similar to that of the
preceding article and is left as an exercise for the student.


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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