To view this presentation, you'll need to enable Flash.

Show me how

After you enable Flash, refresh this webpage and the presentation should play.

Loading...

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

The Adobe Flash plugin is needed to view this content

View by Category

Presentations

Products
Sold on our sister site CrystalGraphics.com

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

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)

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Presentation Removal Request Contact Us Send Us Feedback

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Introduction to Conics" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!