Conic Sections

Four conic sections:Four conic sections:

• Circles• Parabolas• Ellipses• Hyperbolas

A plane intersecting a double cone will generate:

Parabola

Circle Ellipse Hyperbola

Circle

©National Science Foundation

• The Standard Form of a circle with a center at (h,k) and a radius, r

222 )()( rkyhx

                                                                  

                        

                                                                              

center (3,3)radius = 2

Circle

Parabola

© Art Mayoff © Long Island Fountain Company

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.

Demo

Why is the focus so important?

the parabola opens: UP if p > 0, DOWN if p < 0

Vertex: (h, k)Focus: (h, k+p)Directrix: y = k - pAxis of symmetry: x = h

Parabola with vertical axis of symmetryStandard equation: (x – h)2 = 4p(y – k)

p ≠ 0

Parabola with horizontal axis of symmetry

the parabola opens: RIGHT if p > 0, LEFT if p < 0 Vertex: (h, k)Focus: (h + p, k)Directrix: x = h – pAxis of symmetry: y = k

Standard equation:(y – k)2 = 4p(x – h)

p ≠ 0

Practice – Parabola

• 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).

Ellipse

© Jill Britton, September 25, 2003

•Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point.

Ellipse• The ellipse is the set of all points in the plane for

which the sum of distances from two fixed points (foci) is a positive constant.

Parts of an Ellipse:

• 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

Why are the foci of the ellipse important?

• St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.

© 1994-2004 Kevin Matthews and Artifice, Inc. All Rights Reserved.

Why are the foci of the ellipse important?

• The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

Ellipse with horizontal major axis:

Center is (h, k).Length of major axis is 2a.Length of minor axis is 2b.Distance between center and either focus is c with c2 = a2– b2, a > b > 0.

Ellipse with vertical major axis:

Center is (h, k).Length of major axis is 2a.Length of minor axis is 2b.Distance between center and either focus is c with c2 = a2– b2, a > b > 0.

Practice – Ellipse:• Graph 4x 2 + 9y2 = 4

• Find the vertices and foci of an ellipse - sketch the graph:

4x2 + 9y2 – 8x + 36y + 4 = 01. put in standard form2. find center, vertices, and foci

• Write the equation of the ellipse with:center at (4, -2), the foci are (4, 1) and (4, -5) and the

length of the minor axis is 10.

Hyperbola

The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

© Jill Britton, September 25, 2003

Where are the Hyperbolas?

• A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in

its path.

© Jill Britton, September 25, 2003

Hyperbola

• The hyperbola is the set of all points in a plane for which the difference between the distances from two fixed points (foci) is a positive constant.

Differs from an Ellipse whose sum of the distances was a constant.

Parts of the hyperbola:

• Transverse axis • Conjugate axis • Vertices• Foci (will be on the transverse axis)• Center• Asymptotes

Hyperbola with horizontal transverse axis

Center is (h, k).Distance between the vertices is 2a.Distance between the foci is 2c.c2 = a2 + b2

Eccentricity of the hyperbola:

• Eccentricity e = c/a since c > a , e >1• As the eccentricity gets larger the graph becomes

wider and wider

Hyperbola with vertical transverse axis

Center is (h, k).Distance between the vertices is 2b.Distance between the foci is 2c.c2 = a2 + b2

+-

Practice – Hyperbola:

• Graph

• Write in standard form:9y2 – 25x2 = 2254x2 –25y2 +16x +50y –109 = 0

• Write the equation of the hyperbolas: Vertices (0, 2) and (0, -2) Foci (0, 3) and (0, -3)

Vertices (-1, 5) and (-1, -1) Foci (-1, 7) and (-1, 3)

13625

22

xy

1144

3

25

6 22

yx

Conclusion:

2 2 0

where A, B, and C are not all zero.

Ax Bxy Cy Dx Ey F

Most General Equation of a Conic Section:

Which Conic is it?

• Parabola: A = 0 OR C = 0

• Circle: A = C• Ellipse: A ≠ C, but both have the same sign

• Hyperbola: A and C have Different signs

Example: State the type of conic and write it in the standard form of that conic

• Conic: A and C same sign, but A ≠ C• ELLIPSE

• Standard Form:

2 22 8 12 4 0x y x y

2 2

2 2

( ) ( )1

x h y k

a b

2 2( 8 __) 2( 6 __) 4 __ __x x y y 2 2( 8 16) 2( 6 9) 4 16 18x x y y

2 2( 4) 2( 3) 38x y 2 2( 4) ( 3)

138 19

x y

Example: State the type of conic and write it in the standard form of that conic

• Conic: A = 0• PARABOLA

• Standard Form:

26 2 24 10 0y x y

2( )x a y k h 22 6 24 10x y y

23( 4 __) 5 ( 3)(__)x y y 23( 4 4) 15 2x y y

23( 2) 17x y

(y – k)2 = 4p(x – h)

Example: State the type of conic and write it in the standard form of that conic

• Conic: A =C• CIRCLE

• Standard Form:

2 22 2 8 12 4 0x y x y

2 2 2( ) ( )x h y k r

2 22( 4 __) 2( 6 __) 4 __ __x x y y 2 22( 4 4) 2( 6 9) 4 8 18x x y y

2 22( 2) 2( 3) 30x y 2 2( 2) ( 3) 15x y

**Divide all by 2!**

HW assignment - Identify, write in standard form and graph each of the following:1) 4x2 + 9y2-16x - 36y -16 = 0

2) 2x2 +3y - 8x + 2 =0

3) 5x - 4y2 - 24 -11=0

4) 9x2 - 25y2 - 18x +50y = 0

5) 2x2 + 2y2 = 10

6) (x+1)2 + (y- 4) 2 = (x + 3)2

Additional images