Loading...

PPT – 10'3 Conics The Parabola PowerPoint presentation | free to view - id: f505d-NmI2Z

The Adobe Flash plugin is needed to view this content

10.3 ConicsThe Parabola

Definition of a Parabola

- A Parabola is the set of all points in a plane

that are equidistant from a fixed line (the

directrix) and a fixed point (the focus) that is

not on the line.

Animation of Definition

Applications of a Parabola

- Some comets shoot through our solar system along

parabolic paths with the Sun at the focus. Notice

how the comet speeds up as it gets closer to the

Sun... It's a gravity thing! - Look at the comet's tail. The solar wind always

blows the tail away from the Sun. By the way, I

highly recommend that you take an astronomy class

in college. You'll love it!)

Parabolic applications

Parabolic applications

- Automobile Headlights
- An automobile headlight is another

example of a Paraboloid of Revolution --taking a

parabola and rotating it about its axis of

symmetry. The smooth inner surface of the

headlight is a glass reflector upon which bright

aluminum has been deposited. This part is a

powerful reflector. - A parabolic reflector has the property

that if a light source is placed at the focus of

the reflector, the light rays will reflect from

the mirror as rays parallel to the axis.

This is used in auto headlights to give an

intense concentrated beam of light.

Parabolic applications

- The principle of the parabolic reflector may have

been discovered in the 3rd century BC by the

geometer Archimedes, who, according to a legend

of debatable veracity constructed parabolic

mirrors to defend Syracuse against the Roman

fleet, by concentrating the sun's rays to set

fire to the decks of the Roman ships.

Le Four Solaire at Font-Romeur There is a

reflector in the Pyrenees Mountains that is 8

stories high. It cost two million dollars to

build and it took ten years to build it. It is

made of 9,000 mirrors arranged in a parabolic

mirror. It can reach 6,000 degrees Fahrenheit

just from the Sun!

Parabolic applications

Gallileo was the first to show that the path an

object thrown in space is a parabola.

Parabolic applications

The cables that act as suspension are parabolas.

Standard Forms of the Parabola

- The standard form of the equation of a parabola

with vertex at the origin is - y2 4px or x2 4py.
- The graph illustrates that for the equation on

the left, the focus is on the x-axis, which is

the axis of symmetry. For the equation of the

right, the focus is on the y-axis, which is the

axis of symmetry.

y

x

Example 1

- Find the focus and directrix of the parabola

given by

4p 16p 4Focus (0,4) and directrix y -4

Example 1

Find the focus and directrix of the parabola

given by x2 -8y. Then graph the parabola.

The given equation is in the standard form x2

4py, so 4p -8. x2 -8y

Example 1 cont'd.

Find the focus and directrix of the parabola

given by x2 -8y. The graph the parabola.

Because p -2, p lt 0, the parabola opens

downward. Using this value for p, we

obtain Focus (0, p) (0, -2) Directrix

y - p y 2.

To graph x2 -8y, we use test points. If we

assign y a value that makes the right side a

perfect square, it makes it easier. If y -2,

then x2 -8(-2) 16, so x is 4 and 4. The

parabola passes through the points (4, -2) and

(-4, -2).

Example 2

Find the standard form of the equation of a

parabola with focus (5, 0) and directrix x -5.

The focus is (5, 0). implies the focus is on the

x-axis. Use the standard form of the equation

in which x is not squared, y2 4px. We need to

determine the value of p. The focus, located at

(p, 0), is p units from the vertex, (0, 0). then

p 5. We substitute 5 for p into y2 4px y2 4

5x or y2 20x.

Standard form if vertex is not at the origin

(y k)2 4p(x h). (x h)2 4p(y

k).

Example 3

Find the vertex, focus, and directrix of the

parabola given by y2 2y 12x 23 0. Then

graph the parabola.

Vertex ( 2 , -1 )

(y 1)2 -12x 24

p -3

(y 1)2 -12(x 2)

Example 3 cont'd.

(y 1)2 -12(x 2)

Vertex ( 2 , -1 )

y is the squared term and p is negative -3

Directrix is p units from the vertex x 5

Focus pt. is p units from the vertex (-1, -1)

Sample point Let x -1 (y

1)2 36 (y 1) 6 y - 1

6 (-1, 5) and ( -1 -7)

The Latus Rectum and Graphing Parabolas

- The latus rectum of a parabola is a line segment

that passes through its focus, is parallel to its

directrix, and has its endpoints on the parabola. - The length of a parabola's latus rectum is 4p,

where p is the distance from the focus to the

vertex.

Test tomorrowCh 9 - 10.3