CONIC SECTIONS

1
CHAPTER 9

• CONIC SECTIONS

2
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

3
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

4
Graph

• Center is at the origin, horizontal axis6 (left
  endpt (-3,0), right endpt (3,0), vertical axis
  8 (top endpt (0,4), bottom endpt (0,-4))

5
What is c?

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

6
What is the equation of an ellipse, centered at
the origin with a horizontal axis10 and vertical
axis8?
7
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 agtb, the horizontal axis will be the major
  axis and the focal points will be on that axis of
  the ellipse.

8
Graph
9
What is the distance from the center to each
focal point?(c)If the center is at (2,-3), the
foci are atSince the major axis is the
horizontal one, you move c units left right of
the center.
10
9.2 The Hyperbola

• Objectives
• Locate a hyperbolas 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

11
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

12
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.)

13
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.

14
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)

15
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

16
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 ,

17
Graph of examplefrom previous slide
18
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

19
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.

20
Standard form of a parabola centered at the
origin, p distance from the center to the focus

• Opens left (plt0),or right (pgt0)
• Opens up (pgt0) or down (plt0)
• Distance from vertix to directrix -p

21
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

22
Translate the parabola center at (h,k)

• Vertical axis of symmetry
• Horizontal axis of symmetry

23
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
  y1
• Graph, next slide

24
Graph of previous slide example
25
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

26
Identifying a conic without completing the square
(A,C not equal zero)

• Circle, if AC
• Parabola, if AC0
• Ellipse if AC Not equal 0, ACgt0
• Hyperbola, if AClt0

27
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

28
Rotation of Axes

• A conic could be rotated through an angle
• The xy-coordinate system is the standard
  coordinate system. The xy-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

29
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.

30
How do we determine the amount of rotation of the
axes?
31
Identifying a conic section w/o a rotation of axes
32
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

33
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

34
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.

35
9.6 Conic Sections in Polar Coordinates

• Objectives
• Define conics in terms of a focus and a directrix
• Graph the polar equations of conics

36
Focus-Directrix Definitions of the Conic Sections
37
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