Conic Sections

1
Conic Sections

• MAT 182
• Chapter 11

2
Four conic sections
Cone intersecting a plane

• Hyperbolas
• Ellipses
• Parabolas
• Circles (studied in previous chapter)

3
What you will learn

• How to sketch the graph of each conic section.
• How to recognize the equation as a parabola,
  ellipse, hyperbola, or circle.
• How to write the equation for each conic section
  given the appropriate data.

4
Definiton of a parabola

• A parabola is the set of all points in the plane
  that are equidistant from a fixed line
  (directrix) and a fixed point (focus) not on the
  line.
• Graph a parabola using this interactive web site.
• See notes on parabolas.

5
Vertical axis of symmetry

• If x2 4 p y the parabola
  opens
• UP if p gt 0
• DOWN if p lt 0
• Vertex is at (0, 0)
• Focus is at (0, p)
• Directrix is y - p
• axis of symmetry is x 0
•  
•  
•  

6
Translated (vertical axis)

• (x h )2 4p (y - k)
• Vertex (h, k)
• Focus (h, kp)
• Directrix y k - p
• axis of symmetry x h

7
Horizontal Axis of Symmetry

• If y2 4 p x the parabola opens
• RIGHT if p gt 0
• LEFT if p lt 0
• Vertex is at (0, 0)
• Focus is at (p, 0)
• Directrix is x - p
• axis of symmetry is y 0

8
Translated (horizontal axis)

• (y k) 2 4 p (x h)
• Vertex (h, k)
• Focus (h p, k)
• Directrix x h p
• axis of symmetry y k

9
Problems - Parabolas

• Find the focus, vertex and directrix
• 3x 2y2 8y 4 0
• Find the equation in standard form of a parabola
  with directrix x -1 and focus (3, 2).
• Find the equation in standard form of a parabola
  with vertex at the origin and focus (5, 0).

10
Ellipses

• Conic section formed when the plane intersects
  the axis of the cone at angle not 90 degrees.
• Definition set of all points in the plane, the
  sum of whose distances from two fixed points
  (foci) is a positive constant.
• Graph an ellipse using this interactive web site.

11
Ellipse center (0, 0)

• Major axis - longer axis contains foci
• Minor axis - shorter axis
• Semi-axis - ½ the length of axis
• Center - midpoint of major axis
• Vertices - endpoints of the major axis
• Foci - two given points on the major axis

Focus
Center
Focus
12
Equation of Ellipse

• a gt b
• see notes on ellipses

13
Problems

• Graph 4x 2 9y2 4
• Find the vertices and foci of an ellipse sketch
  the graph
• 4x2 9y2 8x 36y 4 0
• put in standard form
• find center, vertices, and foci

14
Write the equation of the ellipse

• Given the center is at (4, -2) the foci are
  (4, 1) and (4, -5) and the length of the minor
  axis is 10.

15
Notes on ellipses

• Whispering gallery
• Surgery ultrasound - elliptical reflector
• Eccentricity of an ellipse
• e c/a
• when e ? 0 ellipse is more circular
• when e ? 1 ellipse is long and thin

16
Hyperbolas

• Definition set of all points in a plane, the
  difference between whose distances from two fixed
  points (foci) is a positive constant.
• Differs from an Ellipse whose sum of the
  distances was a constant.

17
Parts of hyperbola

• Transverse axis (look for the positive sign)
• Conjugate axis
• Vertices
• Foci (will be on the transverse axis)
• Center
• Asymptotes

18
Graph a hyperbola

• see notes on hyperbolas
• Graph
• Graph

19
Put into standard form

• 9y2 25x2 225
• 4x2 25y2 16x 50y 109 0

20
Write the equation of hyperbola

• Vertices (0, 2) and (0, -2)
• Foci (0, 3) and (0, -3)
• Vertices (-1, 5) and (-1, -1)
• Foci (-1, 7) and (-1, 3)
• More Problems

21
Notes for hyperbola

• Eccentricity e c/a since c gt a , e gt1
• As the eccentricity gets larger the graph becomes
  wider and wider
• Hyperbolic curves used in navigation to locate
  ships etc. Use LORAN (Long Range Navigation
  (using system of transmitters)

22
Identify the graphs

• 4x2 9y2-16x - 36y -16 0
• 2x2 3y - 8x 2 0
• 5x - 4y2 - 24 -110
• 9x2 - 25y2 - 18x 50y 0
• 2x2 2y2 10
• (x1)2 (y- 4) 2 (x 3)2

23
Match Conics

• Click here for a matching conic section worksheet.