Coordinate Geometry and Complex Numbers

84 Pages·1984·7.834 MB·English

by  P. S. W. MacIlwaine, C. Plumpton (auth.)

Core Books in Advanced Mathematics
Coordinate Geometry and
Complex Numbers
Core Books in Advanced Mathematics
GeneralEditor:C.PLUMPTON,Moderatorin Mathematics,
University ofLondonSchool ExaminationsDepartment;
formerly ReaderinEngineering Mathematics,
Queen Mary College, University ofLondon.
Titlesavailable
CoordinateGeometryandComplexNumbers
MethodsofTrigonometry
Newton'sLawsandParticleMotion
MechanicsofGroupsofParticles
Proof
Differentiation
Integration
Vectors
CurveSketching
Core Books in Advanced Mathematics
Coordinate
and
Geometry
Complex Numbers
s. w.
P. Macllwaine
formerly Chief Examiner in Advanced Level Mathematics.
University of London School Examinations Department,
and Head of Mathematics. Sutton Valence School.
C. Plumpton
Moderator in Mathematics. University of
London School Examinations Department;
formerly Reader in Engineering Mathematics.
Queen Mary College. University of London.
Macmillan Education
© P.S.W.MacIlwaine and C. Plumpton 1984
Allrightsreserved.Nopart ofthispublication
maybereproducedortransmitted,inanyform or
byanymeans,without permission.
First published 1984
Published by
MacmillanEducationLimited
HoundmillsBasingstoke HampshireRG21 2XS
and London
Associated companies throughouttheworld
Typeset inHong Kong byAscoTrade Typesetting Ltd.
ISBN978-0-333-36253-2 ISBN978-1-349-07012-1 (eBook)
DOI 10.1007/978-1-349-07012-1
Contents
Preface vii
1 Basic results and techniques
Elementaryresults incoordinategeometry;Loci; The equation ofa
circle; Parametricequations
2 Conics 13
Conics; The parabola;The ellipse;The hyperbola;The rectangular
hyperbola;The line-pair through the origin
3 Polar coordinates. matrices and transformations 37
Polarcoordinates; Lociin polarcoordinates; Area ofa sector ofa
polarcurve;Use ofmatrices to represent linear transformations;
Useof the unit square to identifytransformations
4 Complex numbers 51
How numbers have developed;Operations on complex numbers;
Conjugatecomplex numbers;The Arganddiagram; Modulus and
argument; Distanceand direction on the Arganddiagram; Locion
the Arganddiagram; Productand quotient for complex numbers in
modulus-argumentform; De Moivre'sTheorem
Answers 72
Index 76
Contents v
Preface
Advanced levelmathematics syllabusesare once again undergoing changes in
contentand approach following the revolution in the early 1960swhich ledto
the unfortunate dichotomy between 'modem' and 'traditional' mathematics.
The current trend in syllabuses for Advanced level mathematics now being
developed and published by many GCE Boards is towards an integrated
approach, taking the best of the topics and approaches of modem and tradi
tional mathematics, in an attempt to create a realistic examination target
throughsyllabuseswhicharemaximalforexaminingand minimalforteaching.
In addition, resulting from a number of initiatives, core syllabuses are being
developed for Advanced level mathematics consisting of techniques of pure
mathematics as taught in schools and collegesat this level.
The concept ofacorecan beusedinseveralways,oneofwhichismentioned
above, namely the idea ofa core syllabusto which options such as theoretical
mechanics, further pure mathematics and statistics can be added. The books
inthis seriesare core books involvingadifferent useofthecore idea.They are
books on a range of topics, each ofwhich iscentral to the study of Advanced
levelmathematics, which together cover the main areas of any single-subject
mathematics syllabusat Advanced level.
Particularly at times when economic conditions make the problems of
acquiring comprehensive textbooks giving complete syllabus coverage acute,
schools and collegesand individual students can collect as many of the core
books as they need to supplement books they already have, so that the most
recentsyllabusesof,forexample,the London,Cambridge, AEBand1MBGCE
Boardscan becoveredatminimum expense.Alternatively, ofcourse,thewhole
setofcore books givescomplete syllabuscoverage ofsingle-subjectAdvanced
levelmathematics syllabuses.
The aim of each book is to develop a major topic of the single-subject
syllabusesgivingessentialbookwork,workedexamplesandnumerousexercises
arising from the authors' vast experience of examining at this level.Thus, as
well as using the core books in either of the above ways, they are ideal for
supplementing comprehensive textbooks by providing more examples and
exercises,sonecessaryfor the preparation and revisionfor examinations.
In this book, wecover the requirements ofthe non-specialistmathematician
in coordinate geometry and complex algebra in accordance with the core
syllabus of pure mathematics now beingincluded by GCE Examining Boards
at Advanced level and meeting the requirements of the polytechnics and
universities for entrantsto degreecourses in mathematics-related subjects.
Preface vii
Intheuseofcoordinates,the importanceoftechnique, thatisthechoiceofa
suitable method to tackle a problem, has been stressed. The statement and
proofofstandardpropertiesofconicshas beenkept to a minimum, orcovered
by worked examples. While inevitably lacking experience, the student should
try to acquire and appreciate good technique, so that more difficult problems
canbetackled confidently. Onlythemostelementary knowledgeofcoordinates
has been assumed, and important basicresults are listedfor easyreference.
In the section on complex algebra no previous knowledge is assumed; the
intention is to show the usefulness of complex numbers rather than give a
rigorous development of their properties from a set of axioms. On the other
hand, in accordance with modern attitudes, the underlying structure of the
complex field has been indicated so that the student can pursue this aspect
further ifdesired.
Plenty ofexamplesare providedthroughout the book,bothasexercisesand
aspartofthetext; theworked examplessometimesmakecomparisonsbetween
good and bad methods.
P. S.W.MacIlwaine
C. Plumpton
viii Preface
1 Basic results and techniques
1.1 Elementary results in coordinate geometry
IfPI == (XI,YI) and P2== (X2,Y2),weobtain the following results directly from
Fig. 1.1.
DistancePIP (1.1)
2
Y2 - YI
GradientofPIP (1.2)
2
X2 - Xl
Mid-point ofPIP Xl + X2 YI +Y2) (1.3)
2 ( 2 ' 2 .
Equation ofPIP (1.4)
2
==y=mx+ c,
y
Fig. 1.1
Other important results
Straight lineswith gradientsml, m2 are perpendicular
m =-1. (1.5)
¢> lm2
Perpendiculardistanceof(xl,Yl)from ax +by +c= 0is
aXl +bYl +c (1.6)
J(a2 +b2) •
Basic results and techniques 1
+ +
(The expression aX1 bYl c is positive or negative depending on whether
(xl'Yl) ison one side oftheline orthe other; e.g°.,the points(- 2,3) and (1,1°)
are on opposite sides ofthe line 5x - 2y + 1= since 5.(- 2) - 2.3 + 1<
and 5.1- 2. 1+ 1> 0.)
Example 1 Find an equation °of that line through the meet of the lines
3x +Y - 5= 0, 8x - 3y - 19= whichis perpendicularto the line
9x +7y - 4 = 0.
°
Method (i): Solving 3x +y - 5=°0, 8x - 3y - 19= we have x = 2,
y = -1. Gradientof9x +7y - 4 = is - 9/7; hence the gradientofaperpen
dicularline is7/9.
An equation ofthe required line is
y + 1= ~(x - 2)
== 7x - 9y = 23.
Method(ii): Weuse an importantidea incoordinategeometry.Theequation
(3x +y - 5) +A(8x - 3y - 19)= 0,
where Aisany°non-zero constant,is thatofa straightline through the meet of
3x +y - 5= and 8x - 3y - 19= 0, since the coordinates of this point
clearly satisfy the equation for any value of A. The gradient of t°his line is
(3 + 8A)/(3A- 1).Hence the line isperpendicularto 9x + 7y - 4=
3+8A 7
¢>
3A- 1 9
+
9(3 8A)= 7(3A- 1)
¢>
Hence the required line is
°
+
3x y - 5- -t(8x - 3y - 19)=
== 7x - 9y = 23.
Exercise 1.1
1 Find an equation ofthe straight line which passes through the intersection ofthe
lines
5x - 7y +I =0, 3x+9y- 5=0
and whichpassesthroughtheorigin.
2 The points A and B have coordinates (3,0) and (0,- 3).Given that the area ofthe
4t
triangle ABPis units", findtheequationsofthe twolinesonwhichPcan lie.
2 Coordinate Geometryand Complex Numbers