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Introduction to Conic Sections

- A conic section is a curve formed by the

intersection of _________________________

a plane and a double cone.

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History

- Conic sections is one of the oldest math subject

studied. - The conics were discovered by Greek mathematician

Menaechmus (c. 375-325 BC) - Menaechmuss intelligence was highly regarded he

tutored Alexander the Great.

History

- Appollonius (c. 262-190 BC) wrote about conics in

his series of books simply titled Conic

Sections. - Appollonious nickname was the Great Geometer
- He was the first to base the theory of all three

conics on sections of one circular cone. - He is also the one to give the name ellipse,

parabola, and hyperbola.

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Circles

Circles

- The set of all points that are the same distance

from the center.

Standard Equation

CENTER (h, k) RADIUS r (square root)

Example 1

h

r²

k

Center Radius r

)

(

,

k

Example 2

Center ? Radius ?

Ellipses

- Salami is often cut obliquely to obtain

elliptical slices, which are larger.

Ellipses

- Basically, an ellipse is a squished circle

Center (h,k) a major radius, length from center

to edge of circle b minor radius, length from

center to top/bottom of circle

You must square root the denominator

- History
- 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.

Science

- 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).

Properties of Ellipses

- 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.

Example 3

Center (-4 , 5) a 5 b 2

Parabolas

Parabolas

vertex

vertex

pgt0 Opens UP Opens RIGHT plt0 Opens

DOWN Opens LEFT

- One of nature's best approximations to parabolas

is the path of a projectile.

- 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

opens down

What is the vertex? How does it open?

(-2 , 5)

opens right

What is the vertex? How does it open?

(0 , 2)

The Hyperbola

- 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.

Hyperbolas

Hyperbolas

- Looks like two parabolas, back to back.

Center (h , k)

Opens UP and DOWN

Opens LEFT and RIGHT

(h , k)

(h , k)

Hyperbolas Transverse Axis

Hyperbolas - Application

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.

Example 6

Center (-4 , 5) Opens Left and right

What am I?

Name the conic section and its center or vertex.

circle (0,0)

hyperbola (0,0)

parabola vertex (1,-2)

parabola vertex (-2,-3)

circle (2,0)

ellipse (0,0)

hyperbola (1,-2)

circle (-2,-1)

hyperbola (-5,7)

parabola vertex (0,0)

hyperbola (0,1)

ellipse (-5,4)

- Acknowledgements
- http//hotmath.com/hotmath_help/topics/parabolas.h

tml - http//upload.wikimedia.org/wikipedia/commons/8/85

/Hyperbola_(PSF).png - http//www.funwearsports.com/NHL/CAPITALS/WCDomedH

ockeyPuck.gif - Mathwarehouse.com
- http//britton.disted.camosun.bc.ca/jbconics.htm
- schools.paulding.k12.ga.us/.../Introduction_to_Con

ics.ppt