Loading...

PPT – Introduction to Conics PowerPoint presentation | free to download - id: 7d5e9d-ZTEzO

The Adobe Flash plugin is needed to view this content

Introduction to Conic Sections

- A conic section is a curve formed by the

intersection of _________________________

a plane and a double cone.

(No Transcript)

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.

(No Transcript)

Circles

- The set of all points that are the same distance

from the center.

Standard Equation

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

Warm-Up

-h

r²

-k

Center Radius r

(

)

,

k

Example 2

Center ? Radius ?

Warm-Up

When the tardy bell rings Please have out your

homework, pen to check and pencil and be working

on this warm-up in your spiral below yesterday.

1. 2.

Center ? Radius ?

The Ellipse

- 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 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 foci will

be reflected to the other foci.

Foci

Foci

- 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.

- 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.

(No Transcript)

(No Transcript)

- The ability of the ellipse to rebound an object

starting from one focus to the other focus can be

demonstrated with an elliptical pool table. - When a ball is placed at one focus and is thrust

with a cue stick, it will rebound to the other

focus. - If the pool table is live enough, the ball will

continue passing through each focus and rebound

to the other.

Ellipse

- Basically an ellipse is a squished circle

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

Example 3

2

Center (-4 , 5) a 5 b 2

Parabola

vertex

vertex

- Weve talked about this before
- a U-shaped graph

This equation opens left or right

This equation opens up or down

HOW DO YOU TELLLOOK 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.

- 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

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

- What I look liketwo parabolas, back to back.

This equation opens up and down

This equation opens left and right

Have I seen this before? Sort ofonly now we

have a minus sign in the middle

(h , k)

Center (h , k)

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)