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.