Analytic geometry for colleges, universities schools ebook legaltorrents

Equation of Line passing through Two Points... Tangent and Line through Point of Tangency and 85.. Tangent and Line through Point of Tangency and 115.. The point 0, the intersection of t

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BOSTON NEW YORK.

BY LEACH, SIIKWELL, &SANBOKN.

C J. PETERS&SON,

TYPOGRAPHERS AND ELECTROTYPERS.

PEEFACE.

THIS text-book is designed for Colleges,

Universities,

and Technical Schools The aim of the author has been

to prepare a work for beginners, and at the same time to

make it sufficiently comprehensive for the requirements ofthe usual undergraduate course For the methods ofdevelop

ment of the various principles he has drawn largely uponhis

experience inthe class-room In the preparation of the work

all authors, home and foreign, whose works were available,have been freely consulted.

In the first few chapters

elementary examples follow thediscussion of each principle. In the subsequent chapterssetsof examples appear at intervals throughout each chapter,

and are so arranged as to partake both of the nature of areview and anextension of the

preceding principles At theend of each chapter general

examples, involving a more

extended

application of the principlesdeduced, are placed forthe benefit of those who may desire a higher course in thesubject

The author takes pleasure in calling attention to a "Dis

cussion of

Mathematics in

Washington and Lee University, which

appears as the final chapter in this work

He takes pleasure also in

to Prof C S. Venable, LL.D., University of Virginia, toProf William Cain, C.E., University of North Carolina,

and toProf E S. Crawley, B.S., University of Pennsylvania,for assistance rendered in reading and revising manuscript,

and forvaluable suggestions given

E W. NICHOLS.LEXINGTON, YA

January,1893.

1-3. TheCartesianor BilinearSystem Examples 1

4-6. ThePolar System Examples 4

CHAPTER II.

LOCI.

7. Locusof anEquation. TheEquationof aLocus 9

8 Variables. Constants Examples 10

9. RelationshipbetweenaLocus anditsEquation 11 10-16. DiscussionandConstructionof Loci. Examples 11

CHAPTER III.

THE STRAIGHT LINE

19. The Slope Equation Examples 25

20. TheSymmetricalEquation. Examples 29

21. TheNormal Equation 32

22. Perpendicular Distance of a Point from a Line. Ex

23. Equationof Line, AxesOblique. Examples 35

24. General Equation, Ax +B?y+ C = 37

25. EquationofLinepassingthrougha Point. Examples . 38

26. Equation of Line passing through Two Points. Ex

27. Lengthof LinejoiningTwoPoints. Examples 41

PAGES ARTS..

28 Intersection of Two Lines. Examples

29. Ax + By + C + K(Aa5+ BV + e)=

30. Angle between Two Lines. Examples General Ex amples

CHAPTER IV TRANSFORMATION OF CO-ORDINATES ~rv 31. Objectsof Illustration ou 32. From OneSystemto a ParallelSystem Examples 33* Rectangular Systemtoan Oblique System. Rectangular SystemtoAnother SystemalsoRectangular Examples 34 35 Rectangular System to a Polar System From aPolar Systemtoa Rectangular System Examples General Examples

CHAPTER V THE CIRCLE 36, 37. Generationof Circle. Equationof Circle 59

38. General Equation of Circle. Concentric Circles. Ex amples

39. PolarEquationof Circle

40. Supplemental Chords

41. Tangent Sub-tangent

42. Normal Sub-normal ^7

43. General EquationsofTangent and Normal Examples 44. Lengthof Tangent

45, 46. Radical Axis Radical Centre Examples 70

47. Condition that a Straight Line touch a Circle. Slope Equation of Tangent

48. Chordof Contact

49, 50. PoleandPolar

51. Conjugate Diameters Examples GeneralExamples 77 CHAPTER VI THE PARABOLA 52, 53. Generationof Parabola Equation ofParabola Defini tions

54. Constructionof Parabola

CONTENTS. Vil

56. PolarEquationof Parabola 88

57-59. Tangent Sub-tangent Constructionof Tangent . 89

73. Tangents at the Extremities of a Chord Examples

CHAPTER VII

THE ELLIPSE

78. Constructionof Ellipse 109

79. Latus-Rectum Examples Ill

80. PolarEquationof Ellipse

81. Supplemental Chords 11482,83. Tangent Sub-tangent 115

84. Tangent and Line through Point of Tangency and

85. Methodsof constructingTangents 118 86,87. Normal Sub-normal Examples 119

88. NormalbisectsAngle betweenthe Focal Radii 122

89. Conditionthat a Straight Linetouch theEllipse. SlopeEquationofTangent 123

90. Locus of Intersection of Tangent and Perpendicular

91. Locusof Intersection ofPerpendicularTangents . 125

92. Equationof Chordof Contact 125

126

98. Parallelogramona pair of Conjugate Diameters 131

99. Relation between Ordinates of Ellipse and Circles on

100, 107. Constructionof Hyperbola Latus-Rectum 144

108. RelationbetweenEllipseand Hyperbola 146

109. ConjugateHyperbola Examples 146

110. PolarEquationofHyperbola 149

111. Supplemental Chords 150

112,113. Tangent, Sub-tangent 150

114. Tangent and Line through Point of Tangency and

115. MethodofconstructingTangents 151

118. TangentbisectsAngle betweentheFocal Radii . 154

119. Condition that a Straight Line touch the Hyperbola.SlopeEquationof Tangent 155

120. Locus of intersection of Tangent and Perpendicular

125. Conjugate Diameterslie inthesame Quadrant 157

126, 127. Equation of Conjugate Diameter Co-ordinates of

Extremitiesof Conjugate Diameter 157

CONTENTS. ix

132. Tangent Line, Asymptotes being Axes. The Point

of Tangency 164

133 Intercepts of aTangent ontheAsymptotes 165

134. TriangleformedbyaTangent andtheAsymptotes . 165

135. Intercepts of a Chord between Hyperbola and its

Asymptotes Examples GeneralExamples 165

CHAPTER IX

GENERAL EQUATION OF THE SECOND DEGREE.

138 FirstTransformation Signsof Constants 171

EQUATIONS OF THE THIRD DEGREE

148. TheSemi-cubic Parabola 188

149. Duplicationof Cubebyaid of Parabola 190

157. TheCurveof Sines 203

158. TheCurveof Tangents 204

159 The,

ARTS. PAGES

SPIRALS,

161. TheSpiral of Archimedes 208

162. TheHyperbolicSpiral 210

163. TheParabolicSpiral 212

165. TheLogarithmicSpiral. Examples 214

171. Relation between Systems. Transformation of Co

ordinates. Examples 223

CHAPTER II.

THE PLANE

172. Equation of Plane 226

173. NormalEquationofPlane 227

174. Symmetrical Equationof Plane . 229

175. GeneralEquationof Plane 229

176. Traces Intercepts . 230

177. Perpendicular fromPointon Plane 231

178. Plane throughthree Points 232

179. Any Equation between three Variables. Discussion

CHAPTER III.

THE STRAIGHT LINE

180. Equationsof a StraightLine 236

CONTENTS. xi

182. To find where a given Line pierces the Co-ordinate

183. LinethroughOnePoint 239

184. Line throughTwoPoints. Examples 239

188. Angle between Line andPlane 246

191-193. The ConeanditsSections 250

CHAPTER IV

DISCUSSION OF SUHFACES OF THE SECOND OKDER

GeneralEquationoftheSecond Degree involving three

Variables. TransformationsandDiscussion . 259TheEllipsoidandvarieties 262TheHyperboloidofOneSheetandvarieties 265TheHyperboloidof TwoSheetsandvarieties 267TheParaboloidandvarieties 269SurfacesofRevolution Examples 273

PLANE ANALYTIC GEOMETRY.

PAST L

CHAPTElt I.

CO-ORDINATES THE CARTESIAN OR BILINEAR

SYSTEM.

1. THE relative positions of objects are determined by

referring them to some other objects whose positions are

assumed as known Thus we speak of Boston as situated

in latitude north, and longitude west Here the objects to which Boston is referred are the equator and themeridian passing through Greenwich Or, we speakof Bos

ton as being so manymiles north-east ofNewYork Heretheobjects of reference are the meridian of longitude through

New York and New York itself. In the first case it will beobserved, Boston isreferred totwo lines which intersect eachother at right angles, and the position of the cityis located

when we know its distance and direction from each of theselines.

Inlike manner, if we take anypoint suchasPx (Fig 1) inthe plane of the paper, itsposition is fully determinedwhen

we knowits distance anddirection from each of the two lines

X and Y which intersect each other at right angles inthat plane This method of locating pointsis known by the

name of THE CARTESIAN, orBILINEAR SYSTEM The lines of

reference X, Y, are calledCO-ORDINATE AXES, and, whenread separately, are distinguished as the X-AXIS and theY-AXIS The point 0, the intersection of the co-ordinateaxes, is called the ORIGIN OF CO-ORDINATES, or simply theORIGIN.

The lines x and y which measure the distance of thepoint P! from the Y-axis and the X-axis respectively, are

FIG 1.

called the co-ordinates of the point the distance (x) fromtheY-axis beingcalled the abscissa of thepoint, and the distance (yf

) from the X-axis beingcalled the ordinate of thepoint

2. Keferring to Fig 1, we see that there is a point in each

of the four angles formed by the axes which would satisfy

the conditions ofbeing distant x fromthe Y-axis anddistant

y from the X-axis This ambiguity vanishes when we combine the idea of direction with these distances Inthe case

of places on the earths surface this difficulty is overcome by

using the termsnorth, south, east,andwest Inanalyticgeome

try the algebraic symbols-f-and are used toserve the same

All distances measured to the of the Y-axis

CO-ORDINA TES 3

are called positive abscissas; those measured to the left,negative; all distances measured above the X-axis are calledpositive ordinates; all distances below, negative. With thisunderstanding,the co-ordinates of the point Pj. become(V,y);

isabove the X-axis, and tothe left of the Y-axis; the thirdangle is belowthe X-axis, and to the left of the Y-axis; thefourth angle is below the X-axis and to the right of theY-axis

4. The ordinates of two points are each = b; how is

the linejoiningthem situated with referenceto theX-axis?

Ans Parallel, below

5. The commoji abscissa of two points is a; how is theline joining them situated?

6. In what angles are the abscissas of points positive?

In whatnegative?

7. In what angles are the ordinates of points negative?

8. In whatangles do the co-ordinates of points have likesigns? In what angles unlike signs?

9. The base of an equilateral triangle coincides with theX-axis andits vertex is onthe Y-axis atthe distance3belowthe origin; requiredthe co-ordinates of its vertices ?

Ans (i-V12, 0), (0,

-3), (-iV"l2, 0).

10. If a point so moves thatthe ratio of its abscissa toitsordinate is always = 1, what kind of apath will it describe,

Ans A straightline passing through the origin, and making an angle of 45 withthe X-axis

11. Theextremities of alinearethepoints(2, 1),( 1, 2):

construct the line.

12. If the ordinate of a point is =0, on which of theco-ordinate axes must it lie? If the abscissa is = ?

13. Construct the points ( 2, 3), (2, 3), and showthattheline joiningthem is bisectedat

4. Instead of locating a point in a plane by referring it to

two intersecting lines, we may adopt the second of the two methods indicated in Art 1. The point P1? Fig 2, is fullydeterminedwhen we know its distance PI (=r) and direction P! X (= 0) from some given point in some given

line X. If we give all values from to ooto r, and allvalues from to360 to 0, it is easily seenthat the position

of every point ina plane may be located

This methodof locating a point iscalled thePOLARSYSTEM

CO-ORDINA TES. 5The point is called the POLE; the line X, the POLARAxis, orINITIAL LINE; the distance

r, the EADIUS VECTOR;

the angle 0, the DIRECTIONAL or VECTORIAL ANGLE The

distance r and the angle 0, (r, 0), are called the POLAR CO ORDINATES of a point.

5. Inmeasuringangles in this

system, it is agreed (as intrigonometry),to give the positive sign(+) toallanglesmeas-

(0). A few examples will makethis method of

locatingpointsclear.

If r =2 inches and = 45, then

(2, 45) locates a point

PI 2 inches from the pole, and on a line

making an ano-le

of -f45 withthe initial line.

If r= -2 inches and = 45, then

(-2, 45) locates apoint P3 2 inches from the pole, and on a line making an

angle of 45 with the initial line also; but in this case thepoint is on that portion of the boundary line of the angle

which has been produced backwardthrough the pole.

If r= 2 inches and 9 = - 45, then

-45) locates a

pointP4 2 inches fromthe pole, andout on aline lying belowtheinitialline, and makingan angle of 45 with it.

If r= - 2 inches and 6= - 45, then (-2, -45 )locates apoint P2 directly opposite (with respect tothe pole),the point P4 , (2, -45).

6 Whilethe usualmethodin analyticgeometryoi expressing an angle is in degrees, minutes, and seconds (, , "),

We know from geometrythat angles at the centre ot

samecircle are to each otheras the arcs included between

their sides; hence, if and ff be two central angles, we

Hence circular unit r

-. r 6 =arc X circularunit

If = 360, commonmeasure, then arc=2irr

Hence, rX 360 = 2TTrX circularunit

Therefore the equation,

4. What is the unit of circular measure?

/ i

Ans - 2

9. Constructthe line the co-ordinates of whose extremities

10. How isthe line,the co-ordinates of two of its pointsbeing(3, -\

f3, \ situated with reference to the initial

Ans Parallel

Find the rectangular co-ordinates of the following points:

The EQUATION OF A Locus is the algebraic expression ofthelawsubject to whichthe generatrix movesin describing thatlocus.

If we take any point P3, equally distant from the X-axis

andthe Y-axis, and impose the condition thatitshall somove

and is called the Equation ofthe Locus

Theline P^ called the Locus of the Again

if we take the point P4 , equally distant from the axes,and make it so move thatthe ratioof its ordinate to its abscissa

atanypoint of its pathshall be equalto 1, itwilldescribethe lineP4P2 In this casetheequation of the locus is

x and theline P4P2 is the locus of this equation.

8. It will be observed in either of the above cases (the

line P8 PU its ordinate and abscissa while always equal areyetin a constant state of change, and pass through allvalues

from oo, through0,to-|- GO. For this reason y and x arecalled the VARIABLE or GENERAL CO-ORDINATES of the line.

If we consider the point at any particular position in itspath, as at P, its co-ordinates ( x, ?/) are constant invalue, and correspond to this position of the point, and tothispositionalone Thevariable co-ordinates arerepresented

by x and y, and the particular co-ordinates of the moving

point for any definite position of its path by these letterswith a dash or subscript; or by the first letters of thealphabet, or bynumbers. Thus (x, ?/), (xlt yi), (a, ), (2, 2)correspondto someparticular position of the moving point.

2. A point so moves that its ordinate + a quantity a is

always equal to its abscissa a quantityb; required thealgebraic expression ofthe law

Ans y -\- a \x b.

3. The sumof the squares of the ordinate and abscissa of

a moving point is always constant, and = a2

; what is theequation of its path?

Ans x2

+ 2== a2

4. Give in language the laws of which the following arethe algebraic expressions:

form y =x as indeterminate Here we have foundthat thisequation admitsofa geometric interpretation; i.e.,thatitrepresents a straight line passing through the origin of co-ordinates and making an angle of45 with the X-axis We shall

find, as we proceed, that every equation, algebraic ortranscendental, which does not involve more than three variable quan

tities, is susceptible of a geometric interpretation We shall

find,conversely, that geometric forms can be expressed algebraically, and that all the properties of these forms may bededucedfrom their algebraic equivalents

Let us now assume the equations of several loci, and letuslocateand discuss the geometric forms whichthey represent.

10 Locatethe geometric figurewhosealgebraic&wwdentis t+T*

We knowthat the point where this locus cuts theY-axis has

its abscissa x = 0. If, therefore, we make x = inthe equation, we shall find the ordinate of this point. Making thesubstitution we find y =2. Similarly, the point where the

Mak-ing y= in the equation, we find x = f. Drawing nowthe axesand marking on them the points

we will have two points of the requiredlocus Now make xsuccessively equalto

in the equation, and findthe corresponding values of y. For

convenience let us tabulate theresult thus:

Values ofx Corresponding Values ofy

line and it

is, as we shall see hereafter We shallseealsothat every equation of thefirst degree between twovariablesrepresents some straight line. The distances Oa and Ob

which the line cuts off on the co-ordinate axes are calledINTERCEPTS In locating straightlines it is usuallysufficient

to determine these distances, as the line drawnthrough theirextremitieswill be the locus of theequationfrom whichtheirvalues were obtained

13. Which of the following points areon the locus of theequation 3x2

less than 4 (positive or negative) will always give two realvalues for y; that x J-4 will give y = 0,and thatany

value of x greaterthan -j-4 will give imaginaryvalues for y.

Hence the locus does not extend to the right of the Y-axisfarther thanx = -f-4, norto the leftfartherthanx = 4.

Making x =0, we have y = _j_4

= " "

x = 4.

15 Drawingthe axes andconstructing the points,

-4), (4, 0), ( 4, 0), we have four points ofthe locus; i.e.,B, B, A,A!

YB

This might readily have been inferred from the form

of the equation, for we know that the sum of the squares

of the abscissa (OC) and ordinate (CP^ of any point Pj

in the circle is equal to the square of the radius

(OPj)

We might, therefore, have constructed the locus by takingthe originas centre,and describing acirclewith 4 as aradius.NOTE, x = -JL for any assumed value of y, or y = _j_ 0,

forany assumed value of x always indicates atangency. Re

ferring to the figure we see that as xincreases the values of

ydecrease and become -j- when x =4. Drawing the linerepresented by theequation x =4, we find thatit is tangent

to the curve We shall see also as we proceedthat any twocoincident values of either variable arising from an assumed

or given value of the other indicates a point oftangency

12 Constructand discuss the equation

-J-Locating these points andtracing the curve through them,

we have the required locus Referring to the value of ywe

see fromthe double sign that the curve is symmetrical with

The form of the

onlythe second powers of thevariables), showsthat the locus

is also symmetrical with respect to the Y-axis. Looking

FIG 6.

under the radical we see that any value of x between the

limits +4 and 4will give two real values fory; and thatany value beyondthese limits willgive imaginary values for

y. Hencethe locusis entirelyincluded betweentheselimits.This curve,with which we shall havemore to do hereafter,

is called the ELLIPSE

13 Discussandconstruct the equation

every positive value of x will always give real values for y,the locus must extend infinitely inthe direction of the posi

and as of x will rendery

imaginary, the curve can have no point to the left of theY-axis Making x =0, we find y = -j-0; hence the curvepasses through the origin, and is tangent to the Y-axis.

From these datawe easily see that the locus of the equation

is represented bythe figure below

FK; 7.

This curve is called the PARABOLA.

14 Discussand constructthe equation

4z2

9 ?2=36Hence

We see from the form of the equationthat the locus must

be symmetrical with to both axes under

theradical, wesee thatany value of x numerically less than

_j_3 or 3 will render y imaginary. Hence there is nopoint of the locus within these limits We see also that

any value of x greaterthan +3or 3 will always give realvalues for y. The locus therefore extends infinitely in thedirection of both the positive and negative abscissaefrom thelimitsx = i3.

Making x =0, we find y = -J-2 V 1; hence, the curvedoes notcut the Y-axis

Making y = 0, we find x = -j-3; hence, the curve cutsthe X-axisintwopoints (3, 0), ( 3, 0).

FIG 8.

15 We have inthe precedingexamples confined ourselves

to the construction of the loci of RECTANGULAR equations;

axes Let us now assume the POLAR equation

r= 6 (1 - cos 0)

and discussand constructit.

Assuming values for 6, we find their cosines from some

convenient table of Natural Cosines Substituting thesevalues, we find the corresponding values of r.

Values of6 Values of cos Values ofr

assumed angles with the line OX, and lay off on them

the corresponding values of r. Through these points, tra

smooth we have the requiredlocus

FIG 9.

Thiscurve,fromitsheart-likeshape, iscalled the CARDIOID

16 Discuss andconstructthe transcendentalequation

Passing to equivalent numbers we have & =x, when 2is

the base of the systemof logarithms selected

As the base of a system of

logarithms can neverbe negative, we see from the equation that no negative value ofxcansatisfy it. Hence the locus has none of its

pointsto the left

of the Y-axis. On the otherhand, as

every positive value of

xwill give real values for

y, we see that the curve extendsinfinitelymthe direction of the

positiveabscissae

If = then

If x = 0,then

2* = .-. y =log .-. y = co.

The locus, therefore, cuts the X-axis at a units distance onthe positive side, and continually approaches the Y-axis without ever meeting it. It is further evident that whateverbethe base of the system of logarithms, these conditions must

hold truefor allloci whose equations are of the form a? = x.

Values ofx Corresponding Valuesof y

Locating these points, the curve traced through them will

be the requiredlocus

FIG 10.

This curve is called the LOGARITHMIC Curve, its name

from its equation