CHAPTER 9

CONIC SECTIONS

9.1 THE ELLIPSE

• Objectives– Graph ellipses centered at the origin– Write equations of ellipses in standard form– Graph ellipses not centered at the origin– Solve applied problems involving ellipses

Definition of an ellipse

• All points in a plane the sum of whose distances from 2 fixed points (foci) is constant.

• If an ellipse has a center at the origin and the horizontal axis is 2a (distance from center to right end is a) and the vertical axis is 2b (distance from the center to the top is b), the equation of the ellipse is:

12

2

2

2

b

y

a

x

Graph

• Center is at the origin, horizontal axis=6 (left endpt (-3,0), right endpt (3,0), vertical axis = 8 (top endpt (0,4), bottom endpt (0,-4))

1169

22

yx

What is c?

• c is the distance from the center to the focal point

222 bac

What is the equation of an ellipse, centered at the origin with a

horizontal axis=10 and vertical axis=8?

145

)4

164100

)3

1810

)2

11625

)1

22

22

22

22

yx

yx

yx

yx

What if the ellipse is not centered at the origin?

• If it is centered at any point, (h,k), the ellipse is translated. It is moved right “h” units and up “k” units from the origin.

• Consider:

• The center is at (2,-3), the distance from the center to the right & left endpt = 2, the distance to the top & bottom endpt = 1

• Graph on next slide• Since a>b, the horizontal axis will be the major axis

and the focal points will be on that axis of the ellipse.

1)3(4

)2( 22

yx

Graph

What is the distance from the center to

each focal point?(c)

If the center is at (2,-3), the foci are at

Since the major axis is the horizontal one,

you move c units left & right of the center.

3

314222

c

bac

)3,32(),3,32(

9.2 The Hyperbola

• Objectives– Locate a hyperbola’s vertices & foci– Write equations of hyperbolas in standard

form– Graph hyperbolas centered at the origin– Graph hyperbolas not centered at the origin– Solve applied problems involving hyperbolas

Definition of a hyperbola

• The set of all points in a plane such that the difference of the distances to 2 fixed points (foci) is constant.

• Standard form of a hyperbola centered at the origin:

• Opens left & right

• Opens up & down 1

1

2

2

2

2

2

2

2

2

b

x

a

y

b

y

a

x

What do a & b represent?• a is the distance to the vertices of the

hyperbola from the center (along the transverse axis)

• b is the distance from the center along the non-transverse axis that determines the spread of the hyperbola (Make a rectangle around the center, 2a x 2b, and draw 2 diagonals through the box. The 2 diagonals form the oblique asymptotes for the hyperbola.)

Focal Points

• The foci (focal points) are located “inside” the 2 branches of the hyperbola.

• The distance from the center of the hyperbola to the focal point is “c”.

• Move “c” units along the transverse axis (vertical or horizontal) to locate the foci.

• The transverse axis does NOT depend on the magnitude of a & b (as with the ellipse), rather as to which term is positive.

22 bac

Describe the ellipse:

• 1) opens horizontal, vertices at (4,0),(-4,0)• 2) opens vertically, vertices at (0,5),(0,-5)• 3) opens vertically, vertices at (0,4),(0,-4)• 4) opens horizontal, vertices at (5,0),(-5,0)

100254 22 yx

What if the hyperbola is not centered at the origin? (translated)

• A hyperbola with a horizontal transverse axis, centered at (h,k) is of the form:

1)()(

2

2

2

2

b

ky

a

hx

Describe the hyperbola & graph

• Transverse axis is vertical• Centered at (-1,3)• Distance to vertices from center= 2 units (up &

down) (-1,5) & (-1,1)• Asymptotes pass through the (-1,3) with slopes =

2/3, -2/3• Foci units up & down from the center ,

36)1(4)3(9 22 xy

13)133,1(),133,1(

Graph of examplefrom previous slide

9.3 The Parabola

• Objectives

– Graph parabolas with vertices at the origin

– Write equations of parabolas in standard form

– Graph parabolas with vertices not at the origin

– Solve applied problems involving parabolas

Definition of a parabola

• Set of all points in a plane equidistant from a fixed line (directrix) and fixed point (focus), that is not on the line.

• Recall, we have previously worked with parabolas. The graph of a quadratic equation is that of a parabola.

Standard form of a parabola centered at the origin, p = distance

from the center to the focus• Opens left (p<0),or right (p>0)

• Opens up (p>0) or down (p<0)

• Distance from vertix to directrix = -p

pxy 42

pyx 42

Graph and describe • Write in standard form:

• (1/2)y = (4p)y, thus ½ = 4p, p = 1/8• Center (0,0), opens up, focus at (0,1/8)• Directrix: y = -1/8

22xy yx

2

12

Translate the parabola: center at (h,k)

• Vertical axis of symmetry

• Horizontal axis of symmetry

)(4)( 2 kyphx

)(4)( 2 hxpky

If the equation is not in standard form, you may need to complete the square to achieve standard

form.• Find the vertex, focus, directrix & graph

• Vertex (-2,-1), p= -2, focus: (-2,-3), directrix: y=1• Graph, next slide

)1(8884128)2(

1284

1284

2

2

2

yyyx

yxx

yxx

Graph of previous slide example

9.4 Rotation of Axes

• Objectives

– Identify conics without completing the square

– Use rotation of axes formulas

– Write equations of rotated conics in standard form

– Identify conics without rotating axes

Identifying a conic without completing the square (A,C not equal zero)

• Circle, if A=C

• Parabola, if AC=0

• Ellipse if AC Not equal 0, AC>0

• Hyperbola, if AC<0

022 FEyDxCyAx

A Rotated Conic Section

• Can the graph of a conic be rotated from the standard xy-coordinate system?

• YES! How do we know when we have a rotation? When there is an xy term in the general equation of a conic:

022 FEyDxCyBxyAx

Rotation of Axes

• A conic could be rotated through an angle

• The xy-coordinate system is the standard coordinate system. The x’y’-coordinate system is the rotated system (turning the rotated conic into the standard system)

• Coordinates between (x,y) and (x’,y’) for every point are found according to this relationship:

cos'sin'

sin'cos'

yxy

yxx

Expressing equation in standard form, given a rotated axis.

• Given the equations relating (x,y) and (x’,y’), find (x’,y’) given the angle of rotation

• Substitute these expressions in for x in the equation of the rotated conic. The result is an equation (in terms of x’ & y’) that exists IF the equation were in standard position.

How do we determine the amount of rotation of the axes?

B

CA 2cot

Identifying a conic section w/o a rotation of axes

04:

04:/

04:

2

2

2

ACBHyperbola

ACBcircleEllipse

ACBParabola

9.5 Parametric Equations

• Objectives

– Use point plotting to graph plane curves described by parametric equations

– Eliminate the parameter

– Find parametric equations for functions

– Understand the advantages of parametric representations

Plane Curves & Parametric Equations

• Parametric equations: x & y are defined in terms of a 3rd variable, t: f(t)=x, g(t)=y

• Various values can be substituted in for t to produce new values for x & y

• These values can be plotted on the xy-coordinate system to generate a graph of the functions

Given a function in terms of x & y, can you find its representation as

parametric equations?• Begin by allowing one variable (usually x)

to designated as t. Replace x with t in the expression. Now y is stated in terms of t.

• x may be replace with other expressions involving t. The only restriction is that x and the new expression for x (in terms of t) must have the same domain.

9.6 Conic Sections in Polar Coordinates

• Objectives

– Define conics in terms of a focus and a directrix

– Graph the polar equations of conics

Focus-Directrix Definitions of the Conic Sections

hyperbolae

ellipsee

parabolae

tyeccentriciPD

PFe

spoPdirectrixDfocusf

,1

,1

,1

int:,:,:

Polar Equations of Conics

• (r,theta) is a point on the graph of the conic

• e is the eccentricity

• p is the distance between the focus & the directrix

sin1,

cos1 e

epr

e

epr