1. Conic Sections MAT 182 Chapter 11
Conic section material for lectures for chapter 9

2. Four conic sections Cone intersecting a plane Hyperbolas Ellipses
Parabolas
Circles (studied in previous chapter)
Picture in text on page 526 is very effective

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.
Link to web site to see relationship between focus and parabola shape

5. Vertical axis of symmetry
If x2 = 4 p y the parabola opens
UP if p > 0
DOWN if p < 0
Vertex is at (0, 0)
Focus is at (0, p)
Directrix is y = - p
axis of symmetry is x = 0
Project on smartboard and draw parabola on axis

6. Translated (vertical axis)
(x – h )2 = 4p (y - k)
Vertex (h, k)
Focus (h, k+p)
Directrix y = k - p
axis of symmetry x = h
Project on smartboard and draw parabola on axis

7. Horizontal Axis of Symmetry
If y2 = 4 p x the parabola opens
RIGHT if p > 0
LEFT if p < 0
Vertex is at (0, 0)
Focus is at (p, 0)
Directrix is x = - p
axis of symmetry is y = 0
Project on smartboard and draw parabola on axis

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
Project on smartboard and draw parabola on axis

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).
Work as a group

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.
Web site to show relationship between foci and shape (eccentricity) of ellipse.

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
Emphasis the relationship between a, b, and c…..a > c
Focus
Center
Focus

12. Equation of Ellipse a > b see notes on ellipses
Use the “Notes” link - has ellipses with centers not at origin…also ellipses oriented up/down

13. Problems
Graph x 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
Work as a group

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.
Work as a group

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
Ellipse summary – application problems in text

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.
Emphasis on difference vs sum for ellipses

17. Parts of hyperbola Transverse axis (look for the positive sign)
Conjugate axis
Vertices
Foci (will be on the transverse axis)
Center
Asymptotes
Use smartboard to draw/label the parts

18. Graph a hyperbola see notes on hyperbolas Graph
See “notes” link for hyperbolas with centers not at origin and also opening up and down
Emphasize relationship between a, b, c…… c > a
Work as a group

19. Put into standard form 9y2 – 25x2 = 225 4x2 –25y2 +16x +50y –109 = 0
Work as a group

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
Work as a group

21. Notes for hyperbola Eccentricity e = c/a since c > a , e >1
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)
Application problems in text

22. Identify the graphs 4x2 + 9y2-16x - 36y -16 = 0 2x2 +3y - 8x + 2 =0
Work as a group

23. Match Conics Click here for a matching conic section worksheet.
Work as a group