Title: Introduction to Conics
1Introduction to Conic Sections
2- A conic section is a curve formed by the
intersection of _________________________
a plane and a double cone.
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4History
- 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.
5History
- 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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7Circles
- The set of all points that are the same distance
from the center.
Standard Equation
With CENTER (h, k) RADIUS r (square root)
8Warm-Up
-h
r²
-k
Center Radius r
(
)
,
k
9Example 2
Center ? Radius ?
10Warm-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 ?
11The 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.
12- -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.
13- On a far smaller scale, the electrons of an atom
move in an approximately elliptical orbit with
the nucleus at one focus.
14- Any cylinder sliced on an angle will reveal an
ellipse in cross-section - (as seen in the Tycho Brahe Planetarium in
Copenhagen).
15- 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
16- 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.
17- 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.
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20- 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.
21Ellipse
- 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
22Example 3
2
Center (-4 , 5) a 5 b 2
23Parabola
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.
24- 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.
25- 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.
26- 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.
27- 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.
28- 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.
29Example 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)
30The 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.
31Hyperbolas
- 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)
32Example 6
Center (-4 , 5) Opens Left and right
33What am I?
Name the conic section and its center or vertex.
34circle (0,0)
35hyperbola (0,0)
36parabola vertex (1,-2)
37parabola vertex (-2,-3)
38circle (2,0)
39ellipse (0,0)
40hyperbola (1,-2)
41circle (-2,-1)
42hyperbola (-5,7)
43parabola vertex (0,0)
44hyperbola (0,1)
45ellipse (-5,4)