Pre – Calculus

BSMath 5-2

ii

Acknowledgement

Contents

1 Analytic Geometry 1 1.1 Introduction to Conic Sections and Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

1.2 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

1.3 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . .9

1.4 Systems of Nonlinear Equations . . . . . . . . . . . . . . .14

iv CONTENTS

Unit 1

Analytic Geometry

Introduction

The study of the geometry of gures by algebraic representation and

manipulation of equations describing their positions, congurations, and

separations is called Analytic Geometry .Analytic geometry is also

called coordinate geometry since the ob jects are described as n-tuples

of points (where n=2 in the plane and 3 in space) in some coordinate

system.

2 Analytic Geometry

1.1 Introduction to Conic Sections and

Circles

Conics and Circles

Conic sections are the curves which can be derived from taking slices of a

“double-napped” cone. (A double-napped cone, in regular English, is two

cones “nose to nose”, with the one cone balanced perfectly on the other.)

“Section” here is used in a sense similar to that in medicine or science,

where a sample (from a biopsy, for instance) is frozen or suused with

a hardening resin, and then extremely thin slices (“sections”) are shaved

o for viewing under a microscope. If you think of the double-napped

cones as being hollow, the curves we refer to as conic sections are what

results when you section the cones at various angles. A circle is a geometrical shape, and is not of much use in algebra, since

the equation of a circle isn’t a function. But you may need to work with

circle equations in your algebra classes. In “primative” terms, a circle is

the shape formed in the sand by driving a stick (the “center”) into the

sand, putting a loop of string around the center, pulling that loop taut

with another stick, and dragging that second stick through the sand at

the further extent of the loop of string. The resulting gure drawn in the

sand is a circle. In algebraic terms, a circle is the set (or “locus”) of points (x, y) at

some xed distance r from some xed point (h, k). The value of r is called

the “radius” of the circle, and the point (h, k) is called the “center” of

the circle. The “general” equation of a circle is:

x2

+ y2

+ Dx +E y +F = 0

The “center-radius” form of the equation is: (x h)2

+ ( y k)2

= r2

where the h and the k come from the center point (h, k) and the r2 comes

from the radius value r. If the center is at the origin, so ( h; k) = (0 ;0),

then the equation simplies to x2

+ y2

= r2

.

You can convert the “center-radius” form of the circle equation into

the “general” form by multiplying things out and simplifying; you can

convert in the other direction by completing the square. The center-radius form of the circle equation comes directly from the

Distance Formula and the denition of a circle. If the center of a circle

is the point ( h; k) and the radius is length r, then every point (x, y) on

the circle is distance r from the point ( h; k). Plugging this information

1.1 Introduction to Conic Sections and

Circles 3 into the Distance Formula (using r for the distance between the points

and the center), we get:

r= p (

x h)2

+ ( y k)2

r 2

=

p (

x h)2

+ ( y k)2

r 2

= ( x h)2

+ ( y k)2

Properties of Circle

Circles having equal radii are congruent.

Circles having dierent radii are similar.

The central angle which intercepts an arc is the double of any inscribed

angle that intercepts the same arc.

The radius perpendicular to a chord bisects the chord.

The chords equidistant from the center are equal in length.

A tangent to a circle is at a right angle to the radius at the point of

contact.

Two tangents drawn on a circle from a point outside are equal in length.

The angle subtended at the center of a circle by its circumference is

equal to four right angles.

The circumference of two dierent circles is proportional to their cor-

responding radii.

Arcs of the same circle are proportional to their corresponding angles.

Radii of the same circle or equal circles are equal.

Equal chords have equal circumferences.

The diameter of a circle is the longest chord.

4 Analytic Geometry

1.2 Parabola

A parabola (plural “parabolas”; Gray 1997, p. 45)

is the set of all points in the plane equidistant from

a given line (the conic section directrix) and a given

point not on the line (the focus). The focal param-

eter (i.e., the distance between the directrix and fo-

cus) is therefore given by , where is the distance from

the vertex to the directrix or focus. The surface of

revolution obtained by rotating a parabola about its

axis of symmetry is called a paraboloid. The parabola was studied by Menaechmus in an

attempt to achieve cube duplication. Menaechmus

solved the problem by nding the intersection of

the two parabolas and . Euclid wrote about the

parabola, and it was given its present name by Apol-

lonius. Pascal considered the parabola as a pro-

jection of a circle, and Galileo showed that pro jec-

tiles falling under uniform gravity follow parabolic

paths. Gregory and Newton considered the cata-

caustic properties of a parabola that bring parallel

rays of light to a focus (MacTutor Archive), as illus-

trated above. For a parabola opening to the right

with vertex at (0, 0), the equation in Cartesian co-

ordinates is:

p (

x a)2

+ y2

= x+ a

( x a)2

+ y2

= ( x+ a)2

x 2

2ax +a2

= x2

+ 2 ax+a2

y 2

= 4 ax

The quantity 4 ais known as the lactus rectum. If

the vertex is at ( x

0; y

0) instead of (0

;0), the equation

of the parabola is:

(y y

0)2

= 4 a(x x

0)

.

1.2 Parabola 5

Three points uniquely determine one parabola

with directrix parallel to the x-axis and one with

directrix parallel to the y-axis. If these parabolas

pass through the three points ( x

1; y

1), (

x

2; y

2), and

( x

3; y

3) they are given by equations:

x

2

x y 1

x 2

1 x

1 y

1 1

x 2

2 x

2 y

2 1

x 2

3 x

3 y

3 1

= 0

and

y

2

x y 1

y 2

1 x

1 y

1 1

y 2

2 x

2 y

2 1

y 2

3 x

3 y

3 1

= 0 In polar coordinates, the equation of a parabola with parameter

aand a

center (0 ;0) is given by

r= 2

a 1 +

cos

(left gure). The equivalence with the Cartesian form can be seen by

setting up a coordinate system ( x0

; y 0

) = ( x a; y ) and plugging in

r = p x

0

2

+ y0

2

and = tan

1

( y

0 x

0) to obtain

p (

x a)2

+ y2

= 2

a 1 +

x

a p

(

x a)2

+ y2

6 Analytic Geometry

Expanding and collecting terms,

a + x+ p (

a x)2

+ y2

= 0

so solving for y2

gives ( }). A set of confocal parabolas is shown in the

gure on the right

In pedal coordinates with thepedal point at thefocus, the equation

is

p2

= ar:

The parabola can be written parametrically as

x= at2

y = 2 at

or

x= t

2 4

a

y = t:

A segment of a parabola is a Lissaious curve A parabola may be generated as the envelope of two concurrent line

segments by connecting opposite points on the two lines (Wells 1991).

1.2 Parabola 7

In the above gure, the lines

S P A; S QB;andP OQ are tangent to the

parabola at points A; B;andO;

respectively. Then S P P A

=QO OP

=

BQ QS

(Wells 1991). Moreover,

the circumcircle of P QS passes through the focusF(Honsberger

1995, p. 47). In addition, the foot of the perpendicular to a tangent

to a parabola from the focusalways lies on the tangent at the vertex

(Honsberger 1995, p.48). Given an arbitrary point

Plocated “outside” a parabola, the tangent

or tangents to the parabola through Pcan be constructed by drawing

thecircle having

P Fas adiameter , where

Fis thefocus .Then locate

the points Aand Bat which the circle cuts thevertical tangent through

V .The points T

A and

T

B (which can collapse to a single point in the

degenerate case) are then the points of tangency of the lines P Aand P B

and the parabola (Wells 1991).

8 Analytic Geometry

Thecurvature ,arc length , andtangential angle are

K (t) = 1 2

a (1 + t2

) 3 2

s (t) = a(tp 1 +

t2

+ sinh

1

t)

(t) = tan

1

t:

Thetangent vector of the parabola is

x(

t) = 1 p

1 +

t2

y (

t) = t p

1 +

t2

The plots below show the normal and tangent vectors to a parabola.

1.3 Hyperbola 9

1.3 Hyperbola

A hyperbola (plural “hyperbolas”; Gray 1997,

p. 45) is aconic sectiondened as thelocusof all

points Pin theplanethe dierence of whose dis-

tances r

1 =

F

1P

and r

2 =

F

2P

from two xed points

(thefoci F

1and

F

2) separated by a distance 2

cis a

givenpositiveconstant k,

r 2

r

1 =

k

(Hillbert and Cohn-Vossen 1999, p. 3). Letting P

fall on the left x-intercept requires that

k = ( c+ a) (c a) = 2 a;

so the constant is given by k= 2 a,i.e., the distance

between the x-intercepts (left gure above). The hy-

perbola has the important property that a ray orig-

inating at afocus F

1 re

ects in such a way that the outgoing path lies

along the line from the otherfocusthrough the point of intersection (right

gure above).

The special case of therectangular hyperbolacorresponding to a hy-

perbola with eccentricity e= p 2 , was rst studied by Menaechmus. Eu-

clid and Aristaeus wrote about the general hyperbola, but only studied

one branch of it. The hyperbola was given its present name by Apol-

lonius, who was the rst to study both branches. Thefocusandconic

section directrixwere considered by Pappus (MacTutor Archive). The

hyperbola is the shape of an orbit of a body on an escape tra jectory (i.e.,

a body with positive energy), such as some comets, about a xed mass,

such as the sun. The hyperbola can be constructed by connecting the free end

Xof a

rigid bar F

1X

, where F

1 is afocusand the otherfocus

F

2 with a string

F 2P X

. As the bar AXis rotated about F

1 and

Pis kept taut against

the bar (i.e., lies on the bar), thelocusof Pis one branch of a hyperbola

(left gure above; Wells 1991). A theorem of Apollonius states that for a

10 Analytic Geometry

line segment tangent to the hyperbola at a point

Tandintersectingthe

asymptotes at points Pand Q, then OP

OQ

is constant, and P T=QT

(right gure above; Wells 1991). Let the point

Pon the hyperbola have Cartesian coordinates( x; y), then

the denition of the hyperbola r

2

r

1 = 2

agives

p (

x c)2

+ y2

p (

x + c)2

+ y2

= 2 a

Rearranging and completing the square gives

x2

(c 2

a2

) a2

y 2

= a2

(c 2

a2

);

and dividing both sides by a2

(c 2

a2

) results in

x 2 a

2

y

2 c

2

a2

= 1

:

By analogy with the denition of theellipse,dene

b2

= c2

a2

;

so the equation for a hyperbola withsemima jor axis aparallel to the

x-axisandsemiminor axis bparallel to they-axisis given by

x2 a

2

y

2 b

2

= 1

:

or, for a center at the point x

0; y

0) instead of (0

;0),

( x x

0)2 a

2

(

y y

0)2 b

2

= 1

:

1.3 Hyperbola 11

Unlike theellipse, no points of the hyperbola actually lie on thesemimi-

nor axisbut rather the ratio b a

determines the vertical scaling of the

hyperbola. Theeccentricity eof the hyperbola (which always satises

e ; 1 ) is then dened as

e= c a

= q 1 +

b

2 a

2

:

In the standard equation of the hyperbola, the center is located at ( x

0; y

0)

, thefociare at x

0

c; y

0) , and the vertices are at

x

0

a; y

0). The so-

calledasymptotes(shown as the dashed lines in the above gures) can be

found by substituting 0 for the 1 on the right side of the general equation

y= b a

(

x x

0) +

y

0;

and therefore haveslopes b a

the special case a= b(the left diagram above) is known as arectangular

hyperbolabecause theasymptotesareperpendicular. The hyperbola can also be dened as thelocusof points whose distance

from thefocus Fis proportional to the horizontal distance from a ver-

tical line Lknown as theconic section directrixwhere the ratio is