Conic Sections
description
Transcript of Conic Sections
Conic Sections
MAT 182
Chapter 11
Four conic sections
Hyperbolas
Ellipses
Parabolas
Circles (studied in previous chapter)
Cone intersecting
a plane
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.
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.
Vertical axis of symmetry
If x2 = 4 p y the parabola opens
UP if p > 0DOWN if p < 0
Vertex is at (0, 0) Focus is at (0, p)
Directrix is y = - p axis of symmetry is x = 0
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
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
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
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).
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.
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
Center FocusFocus
Equation of Ellipse
a > b
see notes on ellipses
1b
y
a
x2
2
2
2
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
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.
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
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.
Parts of hyperbola
Transverse axis (look for the positive sign)
Conjugate axis
Vertices
Foci (will be on the transverse axis)
Center
Asymptotes
Graph a hyperbola
see notes on hyperbolas
Graph
Graph
13625
22
xy
1
144
3
25
6 22
yx
Put into standard form
9y2 – 25x2 = 225
4x2 –25y2 +16x +50y –109 = 0
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
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)
Identify the graphs
4x2 + 9y2-16x - 36y -16 = 0
2x2 +3y - 8x + 2 =0
5x - 4y2 - 24 -11=0
9x2 - 25y2 - 18x +50y = 0
2x2 + 2y2 = 10
(x+1)2 + (y- 4) 2 = (x + 3)2
Match Conics
Click here for a matching conic section worksheet.