Introduction to Conic Sections

A conic section is a curve formed by the intersection of

_________________________a plane and a double cone.

CirclesThe set of all points that are the same distance from the center.

Standard Equation: 222 )()( rkyhx

With CENTER: (h, k)

& RADIUS: r (square root)

(h , k)

r

Example 1

81)8()2( 22 yx

-h-k r²

Center:

Radius: r

),(

9

k (), h 82

Example 2

1)1( 22 yx

Center ?

Radius ?

• Tilt a glass of water and the surface of the liquid acquires an elliptical outline.

• Salami is often cut obliquely to obtain elliptical slices which are larger.

The Ellipse

• The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth, since the circle is the simplest mathematical curve.

• In the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci.

• On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.

• Any cylinder sliced on an angle will reveal an ellipse in cross-section

• (as seen in the Tycho Brahe Planetarium in Copenhagen).

• 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.

• The principle is also used in the construction of "whispering galleries" such as in 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.

EllipseBasically an ellipse is a squished circle

Standard Equation: 1)()( 2

2

2

2

b

ky

a

hx

Center: (h , k)

a: major radius (horizontal), length from center to edge of circle

b: minor radius (vertical), length from center to top/bottom of circle

* You must square root the denominator

(h , k)

ab

Example 3

14

)5(

25

)4( 22

yx

a² b

This must equal 1

Center: (-4 , 5)

a: 5

b: 2

2

ParabolaWe’ve talked about this before…

a U-shaped graph

)(4)( 2 kyphx

Standard Equations:

)(4)( 2 hxpky OR

This equation opens up or down This equation opens left or right

HOW DO YOU TELL…LOOK FOR THE SQUARED VARIABLE

Vertex: (h , k)

•If there is a negative in front of the squared variable, then it opens down or left.

•If there is NOT a negative, then it opens up or right.

vertex

vertex

• One of nature's best known approximations to parabolas is the path taken by a body projected upward, as in the parabolic trajectory of a golf ball.

• The easiest way to visualize the path of a projectile is to observe a waterspout.

• Each molecule of water follows the same path and, therefore, reveals a picture of the curve.

• This discovery by Galileo in the 17th century made it possible for cannoneers to work out the kind of path a cannonball would travel if it were hurtled through the air at a specific angle.

• Parabolas exhibit unusual and useful reflective properties.

• If a light is placed at the focus of a parabolic mirror, the light will be reflected in rays parallel to its axis.

• In this way a straight beam of light is formed.

• It is for this reason that parabolic surfaces are used for headlamp reflectors.

• The bulb is placed at the focus for the high beam and in front of the focus for the low beam.

• The opposite principle is used in the giant mirrors in reflecting telescopes and in antennas used to collect light and radio waves from outer space:

• ...the beam comes toward the parabolic surface and is brought into focus at the focal point.

Example 4

)5()2(12

1 2 yx

What is the vertex? How does it open?(-2 , 5) opens down

Example 52)2(1255 yx

What is the vertex? How does it open?(0 , 2) opens right

• If a right circular cone is intersected by a plane perpendicular to its axis, part of a hyperbola is formed.

• Such an intersection can occur in physical situations as simple as sharpening a pencil that has a polygonal cross section or in the patterns formed on a wall by a lamp shade.

The Hyperbola

HyperbolasWhat I look like…two parabolas, back to back.

Standard Equations:

OR1

)()( 2

2

2

2

b

ky

a

hx1

)()( 2

2

2

2

b

hx

a

ky

Have I seen this before? Sort of…only now we have a minus sign in the middle

This equation opens up and downThis equation opens left and right

Center: (h , k)(h , k)

Example 6

14

)5(

25

)4( 22

yx

Center: (-4 , 5)

Opens: Left and right

Name the conic section and its center or vertex.

2522 yx

122 yx

2)2(12

11 yx

2)2(8

13 xy

4)2( 22 yx

1925

22

yx

116

)2(

4

)1( 22

yx

49)1()2( 22 yx

1)7()5( 22 yx

xy 62

14

)1( 22

xy

15

)5(

17

)4( 22

xy