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

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

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

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

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

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

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

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

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.

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

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

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 Work as a group

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

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

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

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

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

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

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

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

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

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