Essentials of engineering mathematics worked examples and problems 2nd edition by alan jeffrey

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SECOND EDITION

ESSENTIALS OF ENGINEERING MATHEMATICS

Worked Examples and Problems

First published in 1992, Essentials of Engineering Mathematics is a

widely popular reference ideal for self-study, review, and fast answers to specific

questions While retaining the style and content that made the first edition so

successful, the second edition provides even more examples, new material, and

most importantly, an introduction to using two of the most prevalent software

packages in engineering: Maple and MATLAB Specifically, this edition includes:

using symbolic software to perform calculations, explore the properties

of functions and mathematical operations, and generate graphical output

The author includes all of the topics typically covered in first-year undergraduate

engineering mathematics courses, organized into short, easily digestible sections that

make it easy to find any subject of interest Concise, right-to-the-point exposition, a

wealth of examples, and extensive problem sets at the end each chapter—with answers

at the end of the book—combine to make Essentials of Engineering

Mathematics, Second Edition ideal as a supplemental textbook, for self-study,

and as a quick guide to fundamental concepts and techniques

Alan Jeffrey is Emeritus Professor of Engineering Mathematics at the University

of Newcastle upon Tyne, UK

Worked Examples and Problems

SECOND EDITION

ESSENTIALS OF ENGINEERING MATHEMATICS

Worked Examples and Problems

CHAPMAN & HALL/CRC

A CRC Press CompanyBoca Raton London New York Washington, D.C

SECOND EDITION

Alan Jeffrey

ESSENTIALS OF ENGINEERING MATHEMATICS

Taylor & Francis Group

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© 2004 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

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Section 1 Real numbers, inequalities and intervals 1

Section 10 Complex numbers: real and imaginary forms 66

Section 12 Modulus
/argument form of a complex number 81

Section 19 Differentiation of inverse trigonometric functions 133

Section 21 Parametrically defined curves and parametric

Section 26 Properties and applications of differentiability 174

Section 28 Limits and continuity of functions of two

Section 32 Change of variable in partial differentiation 230

Section 33 Antidifferentiation (integration) 238

Section 37 Partial fractions and integration of rational

Section 39 The fundamental theorem of integral calculus

and the evaluation of definite integrals 296

Section 42 Geometrical applications of definite integrals 324Section 43 Centre of mass of a plane lamina (centroid) 333Section 44 Applications of integration to he hydrostatic

Section 50 Taylor’s theorem for functions of two variables:

stationary points and their identification 403

Section 53 Matrices: equality, addition, subtraction,

Section 56 Solution of a system of linear equations:

Section 66 Differentiation and integration of vectors 550Section 67 Dynamics of a particle and the motion of a

Section 68 Scalar and vector fields and the gradient of a

Section 69 Ordinary differential equations: order and

degree, initial and boundary conditions 578

Section 70 First order differential equations solvable by

Section 71 The method of isoclines and Euler’s methods 592

Section 72 Homogeneous and near homogeneous equations 605

Section 74 The first order linear differential equation 617

Section 76 The structure of solutions of linear differential

Section 77 Determining the complementary function for

Section 78 Determining particular integrals of constant

Section 79 Differential equations describing oscillations 649

Section 80 Simultaneous first order linear constant

Section 81 The Laplace transform and transform pairs 662

Section 83 The shift theorems and the Heaviside step

Section 85 The delta function and its use in initial value

Section 86 Enlarging the list of Laplace transform pairs 697

Section 87 Symbolic algebraic manipulation by

Preface to the Second Edition

This book evolved from lectures given in Newcastle over many years, and itpresents the essentials of first year engineering mathematics as simply aspossible It is intended that the book should be suitable both as a text

to supplement a lecture course and also, because it contains a full set ofdetailed solutions to problems, as a book for private study

The success with which the style and content of the first edition was ceived has persuaded me that these features should be preserved when pre-paring this second edition Accordingly, the changes made to the originalmaterial have, in the main, been confined to small amendments designed toimprove the understanding of some basic concepts Typical of these amend-ments to the first edition is the inclusion in Section 26 of some new problemsinvolving the mean value theorem for derivatives, an extension of the account

re-of stationary points re-of functions re-of two variables in Section 50 to includeLagrange multipliers, and the introduction of the concept of the direction field

of a first order differential equation in Section 71, now made possible by theready availability of suitable computer software While making these chan-ges, the opportunity has also been taken to correct some typographical errors.More important, however, is the inclusion of a considerable amount of newmaterial The first is to be found in Section 85, where the reader is introduced

to the delta function and its uses with the Laplace transform when solvinginitial value problems for linear differential equations The second, which isfar more fundamental, is the inclusion of an introductory account in Section

87 of the use of new computer software that is now widely available Thepurpose of this software is to enable a computer to act in some ways like aperson with pencil and paper, because it allows a computer to performsymbolic operations, like differentiation, integration, and matrix algebra,and to give the results in both symbolic and numerical form The two exam-ples of software described here are called MAPLE and MATLAB, each ofwhich names is the registered trademark of a software company quoted inSection 87 In fact MAPLE, which provides the symbolic capabilities ofMATLAB, was used when preparing the new material for this second edition.When symbolic software is available the reader is encouraged to take fulladvantage of it by using it to explore the properties of functions, mathema-tical operations, and differential equations, and also by using its excellentgraphical output to gain a better understanding of the geometrical implica-tions of mathematical results

Alan JeffreyNewcastle upon Tyne

Real numbers, inequalities

The study of the calculus and its many applications depends crucially on theproperties of real numbers The set of all real numbers is often representedsymbolically by writing either R or R If x is a real number this is oftenshown by writing x
R or x
/R Here the symbol
/is to be read ‘belongs to’.The formal mathematical name for this symbol is the set membership relationsymbol There are three different types of real numbers:

1 Positive integers or natural numbers 1, 2, 3,

2 Rational numbers (fractions) of the form p /q with p, q integers with nocommon factor, such as 1/3, 27/5,
/5/16,

3 Irrational numbers or numbers such as
2 which cannot be expressed as arational number

Calculations with real numbers depend on what are called the field axiomsfor real numbers which determine how real numbers may be combined

Field axioms

1 Real numbers commute with respect to the operations of addition andmultiplication, in the sense that if x , y are real numbers, then

xy  yx and xy  yx:

Thus, commutativity means that the order in which real numbers areadded or multiplied is immaterial

2 Real numbers are associative with respect to the operations of additionand multiplication, in the sense that if x , y, z are real numbers, then

x(yz)  (xy)z and x(yz)  (xy)z:

Thus associativity means that the order in which real numbers aregrouped when performing additions or multiplications is immaterial

3 Real numbers are distributive with respect to multiplication, in the sensethat if x , y, z are real numbers, then

x(yz)  xyxz:

Thus the distributivity means that the product of a number and a sum isequal to the sum of the respective products

4 Real numbers 0 and 1 exist, called identity elements with respect toaddition and multiplication, and they have the property that if x is areal number, then

7 If x and y are real numbers, then x/y and xy are also real numbers This

is called the closure axiom, and it says that adding or multiplying two realnumbers can only produce another real number (it cannot produce adifferent type of number */say a complex number)

ORDER PROPERTY OF REAL NUMBERS

There is a natural order amongst the real numbers which always makes itpossible to say which of two different numbers is the larger If x and y aretwo distinct (different) real numbers, with x greater than y, we write

The signs /, B/(respectively read ‘greater than’ and ‘less than’) are calledinequality signs The obvious modifications

x E y and x 0 ymean, respectively, that x is greater than or equal to y and x is less than orequal to y

Extensive use is made of inequalities throughout mathematics and itsmany applications Although they are largely self-evident, the followingelementary inequalities arise sufficiently frequently for it to be worthwhilelisting them for future reference

Elementary inequalities

1 If a /b and c E/d , then a/c /b/d

2 If a /b E/0 and c E/d /0, then ac /bd

3 If k /0 and a /b, then ka /kb ; and if k B/0 and a /b, then ka B/kb

4 If 0 B/a B/b, then a2B/b2; and if a B/b B/0, then a2/b2

5 If a B/0 or a /0, then a2/0

6 If a /b, then
/a B/
/b

7 If a B/0, b /0, then ab B/0, while if a B/0, b B/0, then ab /0

8 If a /0, then 1/a /0, while if a B/0, then 1/a B/0

9 If a /b /0, then 1/b /1/a /0, while if a B/b B/0, then 1/b B/1/a B/0

When working with real numbers it is necessary to have a measure of the

magnitude (size) of a real number without regard to it sign This measure is

provided by the absolute value of the number which is defined as follows

If a is a real number, then its absolute value, written ja j, is defined as

½a½ a if a E 0

a if a B 0:

Thusja j is a non-negative number (i.e., positive or zero, but never negative)

which measures the magnitude of a For example j /25j /25, j7j/7,

j /2pj /2p andj4/3j /4/3

Two important and useful properties of the absolute value are that if a and

b are real numbers,

½ab½ ½a½½b½ and ja

bj½a½

½b½, provided b " 0:

See Example 1.1, which now follows, for a proof of these results

Example 1.1

Prove that if a , k are real numbers:

(i) jka j /jk j jaj;

(i) We consider each case of a and k positive, negative and zero

1 If either k or a (or both) is zero, thenjka j/0 /jk j ja j

2 If k /0, a /0, thenjka j/ka , butjk j ja j/ka , sojka j/jk j jaj

3 If k B/0, a /0, then jka j //ka , but jk j ja j /( /k )a /
/ka , sojka j/jk j jaj The same form of argument shows the result to betrue if k /0 and a B/0

4 If k B/0 and a B/0, thenjka j /ka , whilejk j jaj /(
/k )(
/a ) /ka ,

so thatjka j /jk j ja j The result is proved

(ii) This result follows directly from (i) by setting K/1/k and using the factthatjKa j /jK j ja j

(iii) If a /0, thenja j/a , soja j2

/a2

If a B/0, thenja j//a , andja j2

/(
/a )2/a2

If a /0, thenjaj /0 /a and againja j2

/a2, so the result is proved.(iv) This equation is best solved using the interpretation ofja
/bj as thedistance between a and b, and plotting points on the x -axis We saymore about such geometrical representations after this problem Using(i) we may write

The order property of real numbers allows them to be represented by points

on a straight line To do this, we take a point on the line as an origin 0, a unit

of length to represent the integer 1, and use the convention that distances tothe right of 0 are positive and those to the left are negative Then, for

Fig 1

example, the number 2.9 is represented by the point distant 2.9 length units

to the right of 0, while the number
/1.3 is represented by the point 1.3

length units to the left of 0, as shown in Fig 2 The set of all real numbers R is

called the real line, and the numbers 2.9 and
/1.3 are two particular

numbers belonging to R, so that 2.9
/R and
/1.3
/R As numbers and

points are equivalent in such a representation, the terms ‘number’ and ‘point’

are used interchangeably

Inequalities are useful for identifying intervals on a line, and these are

necessary when working with functions, and elsewhere An interval on the

real line R is the set of all points (numbers) between two specified (end)

points on the real line An interval may be open, closed or half-open,

depend-ing on whether both end points are excluded from the intervals, both are

included, or one is included and the other excluded Graphically, an end

point excluded from an interval is shown as a small circle on the line and one

which is included as a solid dot

Equivalent notations for intervals

Infinity () is a limiting operation, and not a number, so when it occurs

at the end of an interval, by convention this is regarded as an open end

Fig 2

5 Infinite interval

 B x B ; (
, ):

When using the absolute value to define an interval, it is helpful to interpretja
/bj as the distance between the points representing a and b on the real line,with the distance regarded as a non-negative quantity as in Example 1.1 (iv).Example 1.2

Show graphically the intervals

½x
3½ B 1 is equivalent to 3
x B 1, or to 2 B x:

Combining results we have

2 B xB 4:

Method 3

In this method we make use of the fact that ja j2/a2 so that, in

particular, jx
/3j2/(x
/3)2 Squaring both sides of the inequality

and using elementary inequality property 4 we have

(x
3)2B 1 and so x2
6x8 B 0:

As x2
/6x/8 /(x
/2) (x
/4) this is equivalent to

(x
2)(x
4) B0:

For this product to be negative, one factor must be positive and the

other negative This is only possible for x in the interval

2 B xB 4,

so again we arrive at the same inequality

(iv) The inequality 0 B/jx
/3jB/1 says that the distance of x from the fixed

point 3 must be greater than zero and strictly less than 1 Thus this case

only differs from the situation in (iii) earlier by the exclusion of the single

point x /3 from the open interval 2 B/x B/4 Hence the inequality

0 B/jx
/3jB/1 defines the points in the open intervals 2 B/x B/3 and

3 B/x B/4 Graphically, these inequalities correspond to the situation

If (x
/1)(x/2) /0 we may multiply the original inequality by this product

and leave the sign /unchanged to obtain

(x2)2 x(x
1) or x24x4  x2
x,

which is equivalent to

5x 
4, or to x 
4=5:

However, (x
/1)(x/2) will be positive if either x /1 and x /
/2 (both

factors are positive) or if x B/1 and x B/
/2 (both factors are negative)

If, however, (x
/1)(x/2) B/0, after multiplying the original inequality

by this product we must reverse the sign /to B/to obtain

Combining cases (i) and (iv) we see that

Example 1.4

Prove that if a , b are any two real numbers, then

(i) ja/bj 0/ja j/jb j (triangle inequality),

(ii) jja j
/jb jj 0/ja
/bj

(jajjbj)2jaj22jajjbjjbj2,but ab 0/ja j jb j, so

½ab½20 (jajjbj)2,and hence

½ab½ 0 ½a½½b½:

(ii) Writing a/(a
/b )/b, it follows from the triangle inequality that

½a½ 0 ½a
b½½b½,and so

½a½
½b½ 0 ½a
b½:

Also b /(b
/a )/a , so from the triangle inequality we have

½b½ 0 ½b
a½½a½,or

½b
a½ 0 ½a½
½b½:

However,ja
/bj /jb
/aj, so the result shows that

½a
b½ 0 ½a½
½b½ 0 ½a
b½:

Thusjaj
/jb j lies in the interval [ /ja
/bj, ja
/bj] which is equivalent to

Now the largest value ofjx j in the interval
/3 0/x 0/2 is 3, and so

½x3
5x23½ 0 335×323  75and thus we may set M /75

A more careful examination shows the smallest possible value of M to be

PROBLEMS 1

Mark on a line the points satisfying the following inequalities, using a circle

to indicate an end point which is omitted from an interval and a dot toindicate an end point which is included

20 What is the value ofa  ½a½

a ½a½ when (a) a /0 and (b) a B/0 ?

21 Let {a1, a2, , an}, {b1, b2, , bn} and {kn, k2, , kn} be any three

sets of n positive numbers, with m the smallest of the n numbers kiand

M the largest, then

22 Let a and b be any two non-negative numbers (they may be positive or

zero, but not negative) and let p, q be positive integers By considering

the product (/ap
bp)(aq
bq), prove that

p:

24 This problem outlines a proof by contradiction that
2 is irrational

Suppose, if possible, the converse is true and
2 is rational, and thus

can be expressed in the from m /n , with m and n integers with no

common factor By squaring both sides of the expression
2 /m /n ,

show that m and n must have a common factor, thereby contradicting

the original assumption Conclude from the contradiction that
2

cannot be expressed in the form m /n and so is not rational (it is

irrational )

Function, domain and

range 2

A simple and typical example of a function is

y  1sin x for p 0 x0 3p=4:

In this example the function is the rule that says ‘to each number x in theinterval /p 0/x 0/3p /4, associate a number y obtained by first finding sin xand then adding the result to unity’ The interval /p 0/x 0/3p /4 is calledthe domain of definition (domain for short) of the function, and the interval

0 0/y 0/2 over which y ranges for all x in the domain is called the range of thefunction The graph of this function, together with its domain and range, isshown in Fig 4

The general definition of a function is that it is a rule (usually a formula)which assigns to every number in the domain of the function a unique number

in the range of the function

A function is usually denoted by a symbol such as f, an arbitrary number

in its domain by x , often called the independent variable or argument of f,and the corresponding number in its range by y, often called the dependentvariable Thus when we write a general function in the form

y  f (x);

Fig 4

f is the function, x is the independent variable and y is the dependent

variable The domain and range are an essential part of the definition of a

function If the domain is not specified, it is understood to be the largest

interval containing x for which the function is defined

The graph in Fig 5 shows a function which is said to be a many
/one

function, in the sense that to one value of y there correspond many (more

than one) values of x In this case, in the interval a 0/x 0/b the values x1, x2,

., x6of x all correspond to the same value y0 The two graphs in Fig 6

show functions which are said to be one
/one or monotonic functions These

are functions which either increase or decrease steadily in a given interval

The graph in Fig 6(a) shows a monotonic increasing function, and the one

in Fig 6(b) shows a monotonic decreasing function In these cases one x

corresponds to one y and, conversely, one y corresponds to one x

The graph in Fig 7 does not represent a function because to one x there

correspond more than one y This is called a one
/many mapping It can be

represented as a set of functions by dividing it up into several different

monotonic functions, as shown, each with its own domain and range In

this case it may be represented as the three monotonic functions:

Fig 5

Fig 6

1 y /y1(x ), monotonic increasing with domain a 0/x 0/c and range

If we regard x1/2as denoting the positive square root, Fig 8 is seen to bedescribed by the two monotonic functions:

1 y/1/x1/2, monotonic increasing with domain x E/0 and range

Notice that yand y
both have the same domain, but different ranges, so

they are different (monotonic) functions

Example 2.1

Find the largest possible domain and corresponding range for each of the

following functions:

(i) y /sin x ;

(ii) y/(3 /x )1/2, where the positive square root is taken;

(iii) y//(1 /x2)1/2, where the positive square root is taken;

(iv) y/j1 /xj1/2

, where the positive square root is taken

Solution

(i) sin x is defined for all x (it is periodic with period 2p ) so the largest

possible domain is the infinite interval / B/x B/, and the range is

then the closed interval /1 0/y 0/1

(ii) (3/x )1/2is only real when 3 /x E/0 Thus the largest possible domain

is the semi-infinite interval x 0/3, and the corresponding range is then

the semi-infinite interval 0 0/y B/

(iii) (1/x2)1/2is only real when x20/1, corresponding to /1 0/x 0/1 Thus

the largest possible domain is the closed interval
/1 0/x 0/1, from which

it follows that /(1 /x2)1/2 then has for its range the closed interval

/1 0/y 0/0

(iv) j1 /xj E/0 for all x , soj1/xj1/2 is defined for all x Thus the largest

possible domain is the infinite interval / B/x B/, and the

corresponding range is then the semi-infinite interval y E/0 m

Clarify the following relationships as one
/one functions, many
/onefunctions or one
/many mappings.

}, where the positive square root is to be taken

22 y /sin {j1 /xj1/2}, where the positive square root is to be taken

A function f (x ) is said to be bounded below by m and bounded above by M for

/a 5 x 5 b if finite numbers m and M can be found such that/m B f (x) B M,where the numbers a and b may be finite or infinite If m is the largestnumber for which this is true, then m is called the greatest lower bound orthe infimum of f (x ) on the interval If M is the smallest number for which this

is true, then M is called the smallest upper bound or the supremum of f (x ) onthe interval A function f (x ) for which no bounds can be found is said to beunbounded Thus sin x is bounded above by 3 and bounded below by/2 forall x , but its greatest lower bound is /1 and its smallest upper bound is 1,while x3is unbounded for all x , but x2has a greatest lower bound of 0 for all

x , but is unbounded above

23 The function 1/[(x/2)(x /3)] is bounded for /1 5/x 5/1, butunbounded for /3 5/x 5/4 Give an example of your own with similarproperties

24 The function f (x ) /4 /x2that has a smallest upper bound 4 for all x ,but is unbounded below Give an example of your own with similarproperties

Basic coordinate geometry 3

The most common graphical representation of a function involves the use ofrectangular Cartesian coordinates These involve two mutually perpendicularaxes on each of which (unless otherwise stated) the same length scale is used

to represent real numbers The horizontal axis is the x -axis, with positive x tothe right of the point of intersection of the two axes which is taken as theorigin, and negative x to the left The vertical axis is the y -axis, with positive

y above the origin and negative y below it

A typical point P in the (x , y )-plane shown in Fig 9 is identified by its

x -coordinate a and its y -coordinate b, with a the number of length units P isdistant from the y -axis, and b the number of length units P is distant fromthe x -axis, with due regard to sign Thus Q is the point (2, 1) and R is thepoint ( /3, /2) The number pair (a , b ) is called an ordered pair because theorder in which a and b appear is important Interchanging a and b in theordered pair (a , b ) to give (b, a ) changes the point represented by thisnotation

The distance AB between points A (x1, y1) and B (x2, x2) in Fig 9 is thelength of the straight line AB so, by Pythagoras’ theorem,

Fig 9

(AB)2 (CB)2(AC)2

jx2x1j2jy2y1j2and hence

is the dashed line in the figure parallel to the x -axis and passing through thepoint c on the y -axis Lines parallel to the y -axis are of the form x /c , where

c is a constant

A straight line is completely specified if:

1 m and c are given,

2 m is given together with a point P (x1, y1) on the line,

3 two points P (x1, y1) and Q (x2, y2) on the line are given

Fig 10

Case 1

If m and c are given, the equation of the straight line

y  mxccan be written down immediately

Case 2

If m and P (x1, y1) are given, only the constant c in the equation

y  mxcneeds to be determined As the line passes through the point P (x1, y1), it

follows that y /y1when x /x1, so substituting into the equation we have

y1 mx1c, or c  y1mx1:Thus the equation of the straight line becomes

y  mxy1mx1or

y  m(xx1)y1:Case 3

If the line passes through P (x1, y1) and Q (x2, y2), the gradient m of the line is

m y2 y1

x2 x1:Substituting this value for m into y /mx/c and using the fact that the line

passes through P (x1, y1) (or Q (x2, y2)) determines c and leads to the

equa-tion of the line in the form

SHIFT OF ORIGIN

The change of variable

X  xa, Y  ybrepresents a shift of every point in the (x , y )-plane by an amount a in the

x -direction and b in the y -direction This is called a shift of origin without

scaling or rotation because distances between points are unaltered and no

rotation occurs Thus any graph of a function in the (x , y )-plane is simply

shifted (translated) without scaling or rotation to the (X , Y )-plane in whichthe origin corresponds to the point (a , b ) in the (x , y )-plane.

An important application of this simple transformation is the proof of theresult that two straight lines

y  m1xc1 and y  m2xc2are orthogonal (mutually perpendicular) if

m1m21:

To prove this result, it will suffice for us to consider the two orthogonallines L1(y /m1x ) and L2(y /m2x ) through the origin in Fig 11 This followsbecause if they intersect at the point (a , b ) instead of at the origin, a change

of variable will reduce them to this case The line x /1 intersects L1at (1, m1)and L2at (1, m2), so by Pythagoras’ theorem

(AB)2 (OA)2(OB)2thus

(11)2(m2m1)2 (1m2

1)(1m2

2),and after simplification this reduces to

m1m21:

Example 3.1

Find the equation of the straight line y /mx/c such that:

(i) m /2 and the line passes through the point (1, /3);

(ii) the line passes through the points (/1, 2) and (3, 4);

(iii) it is the line through the point (2, /5) orthogonal to y /3x /11

Fig 11

(i) This is case 2 in which m /2, x1/1 and y1//3 Thus the equation of

the line is

y  2(x1)3 or y  2x5:

(ii) This is case 3 in which x1//1, y1/2, x2/3 and y2/4 Thus the

equation of the line is

y  2

 4  2

3  (1)

(x(1)) or y 1

2(x5):

(iii) The gradient of the given line is 3, so the gradient of the orthogonal

straight line must be /1/3 (so that m1m2//1) Thus the required line

The equation of a straight line is an example of a polynomial in x of degree

1, also called a linear function of x A polynomial Pn(x ) in x is an expression

of the form

Pn(x)  a0xna1xn1a2xn2 .an1xan,

in which the numbers a0, a1, , an are called the coefficients of the

polynomial, and the number n (the highest power of x in Pn(x )) is called

the degree of the polynomial A polynomial is defined for all x

When n is small, the corresponding polynomials are named as follows:

n /0: a constant function (degree zero);

n /1: a linear polynomial (degree 1);

n /2: a quadratic polynomial (degree 2);

n /3: a cubic polynomial (degree 3);

n /4: a quartic polynomial (degree 4);

n /5: a quintic polynomial (degree 5)

THE CIRCLE

The circle of radius R with its centre at the point (a , b ) shown in Fig 12 has

the equation

(xa)2(yb)2 R2:This is called the standard form of the equation of the circle, and the equa-

tion is derived by applying Pythagoras’ theorem to the triangle ABP where

AB /x /a, PB /y /b and AP /R

An equation of the form

The ellipse is a symmetrical closed curve of the form shown in Fig 13, and it

is characterized geometrically by the fact that the sum of the distances fromtwo points F1and F2called the foci to any point P on the ellipse is a constant,so

d1d2 const:

The longest chord AB of length 2a is called the major axis of the ellipseand the shortest chord CD, which is perpendicular to AB, is called the minoraxis and it is of length 2b, with a
/b The points A and B are called thevertices of the ellipse and point Q the centre of the ellipse

The number

e  c=a,where c2/a2/b2is called the eccentricity of the ellipse The eccentricity issuch that 0 0/e B/1, and when e is small the ellipse is nearly circular, but when

it is close to 1 the ellipse is very elongated

Fig 12

The standard equation of an ellipse with its major axis horizontal and its

centre at the point (a , b ) is

(x  a)2

a2 (y  b)

2

b2  1, with a
b:

The corresponding form of the standard equation of an ellipse when its

major axis is vertical is

The hyperbola is the curve shown in Fig 14, and it is characterized

geometri-cally by the fact that every point on the hyperbola is such that the difference

of its distances from two fixed points F1and F2called the foci is a constant

The line on which F1and F2lie is called the transverse axis of the hyperbola,

and the line perpendicular to the transverse axis which passes through the

mid-point Q of F1F2is called the directrix The point Q is called the centre,

and the points A and B the vertices of the hyperbola The distance of the

vertices from either side of the centre Q is a

The standard form of the equation of the hyperbola with its centre at (a , b )

and its transverse axis horizontal is

Similarly, the standard form of the equation of the hyperbola with its centre

at (a , b ) and its transverse axis vertical is

(y  b)2

a2 (x  a)

2

b2  1,The distance c of the foci from either side of the centre Q is given by

c2 a2b2:

A straight line tangent to a curve at infinity is called an asymptote Theasymptotes to the hyperbola with its centre at (a , b ) and its transverse axishorizontal are

Fig 14

Fig 15

y  b9b

a(xa);

the asymptotes to the hyperbola with its centre at (a , b ) and its transverse

axis vertical are

y  b9a

b(xa):

These asymptotes are shown in Fig 15

THE PARABOLA

The parabola is a curve of the type shown in Fig 16 It is characterized

geometrically by the fact that points on a parabola are such that their

distance from a fixed point F, called the focus, equals their perpendicular

distance from a fixed straight line called the directrix The point A on the

parabola closest to the directrix is called the vertex, and the line through A

perpendicular to the directrix is called axis

The standard form of the equation of a parabola with its axis vertical, its

vertex at the point (a , b ), its focus at the point (a , b/a ) and its directrix

along the line y /b/a is

(xa)2 4a(yb):

The vertex of this parabola will be at the bottom (concave-up) if a
/0 and at

the top (concave-down) if a B/0 The corresponding standard form of the

equation of a parabola with its axis horizontal, its vertex at the point (a , b ),

its focus at the point (a/a , b ) and its directrix along the line x /a/a is

(yb)2 4a(xa):

The vertex of this parabola will lie to the left (concave to the right) if a
/0 and

Fig 16

to the right (concave to the left) if a B/0 Typical examples of these parabolasare shown in Fig 17.

Example 3.2

(i) Show that the equation

x2y24x6y9  0represents a circle, and find its centre and radius

(ii) Find the centre and semi-major and semi-minor axes of the ellipse withthe equation

(ii) To find the centre at point (a , b ) and the positive constants a and b, we

start by expanding the standard equation for an ellipse with its major

a2

b2



a2 0:

If it turns out that the major axis of the ellipse in question is vertical we

will find that b
/a

If necessary, we now rewrite the given equation in the same form as

the reference equation, making the coefficient of x2equal to 1 We will

then compare corresponding coefficients in both the reference equation

and the given equation

As the coefficient of x2is 4, we divide the equation by 4 to obtain

14

By convention, the semi-major axis is greater than the semi-minor axis,

so the equation is seen to be in the second standard form, corresponding

to the case in which the major axis is vertical Thus the semi-major axis

is of length 4 and the semi-minor axis is of length 2

(iii) We start, as in (ii), by expanding the standard equation of a hyperbolawith its axis horizontal

and after combining this result with (E), we see that b2/9 The signs of

a2and b2are the same, so the axis of the hyperbola is horizontal As, by

convention, a and b are positive, it follows that a /2 and b /3 The

standard form of the equation is thus

(x  1)2

(y  1)2

9  1,and substituting for a and b in the equations for the asymptotes to a

hyperbola with its axis horizontal gives

y 193

2(x1):

(iv) A parabola with its axis vertical will contain a term in x2, and one with

its axis horizontal a term in y2 As the equation in question is quadratic

in x the axis must be vertical

Expanding the standard form of the equation of a parabola with its

From (I) we find a /3, from (J) a /2, and from (K) b /1 So the

standard form of the equation of the parabola is

(x3)2 8(y1):

As a /3 and b /1 the vertex is at (3, 1), and as a /3 the focus is at (3, 4)

The parabola is concave-up because the axis is vertical and a  3
0: m

THE ASYMPTOTE

As already stated, an asymptote is a straight line which is tangent to a curve

at infinity Not all curves have asymptotes, but some of the simplest ones to

find belong to rational functions