- By
**talen** - Follow User

- 439 Views
- Updated On :

Conic Sections. Hyperbolas. Definition. The conic section formed by a plane which intersects both of the right conical surfaces Formed when or when the plane is parallel to the axis of the cone . Definition. A hyperbola is the set of all points in the plane where

Related searches for Conic Sections

Download Presentation
## PowerPoint Slideshow about 'Conic Sections' - talen

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Conic Sections

### Conic Sections

Hyperbolas

Definition

- The conic section formed by a plane which intersects both of the right conical surfaces
- Formed whenor when the plane isparallel to the axis of the cone

Definition

- A hyperbola is the set of all points in the plane where
- The difference between the distances
- From two fixed points (foci)
- Is a constant

Geogebra

Demonstration

Experimenting with Definition

- Turn on Explore geometric definition. A purple point will appear on the hyperbola, along with two line segments labeled L1 and L2. Drag the purple point around the hyperbola.
- Do the lengths of L1 and L2 change?
- What do you notice about the absolute value of the differences of the lengths?
- How do these observations relate to the geometric definition of a hyperbola.

- Observe the values of L1, L2, and the difference | L1 − L2 | as you vary the values of a and b.
- How is the difference | L1 − L2 | related to the values of a and/or b? (Hint: Think about multiples.)
- Determine the difference | L1 − L2 | for a hyperbola where a = 3 and b = 4. Use the Gizmo to check your answer.

Elements of An Ellipse

- Transverse axis
- Line joining the intercepts

- Conjugate axis
- Passes throughcenter, perpendicularto transverse axis

- Vertices
- Points where hyperbola intersects transverse axis

Elements of An Ellipse

- Transverse Axis
- Length = 2a

- Foci
- Location (-c, 0), (c, 0)

- Asymptotes
- Experiment with Pythagorean relationship

Equations of An Ellipse

- Given equations of ellipse
- Centered at origin
- Opening right and left
- Equations of asymptotes

- Opening up and down
- Equations of asymptotes

Try It Out

- Find
- The center
- Vertices
- Foci
- Asymptotes

More Trials and Tribulations

- Find the equation in standard form of the hyperbola that satisfies the stated conditions.
- Vertices at (0, 2) and (0, -2), foci (0, 3) and (0, -3)
- Foci (1, -2) and (7, -2)slope of an asymptote = 5/4

Assignment

- Hyperbola A
- Exercise set 6.3
- Exercises 1 – 25 oddand 33 – 45 odd

The Return of the Hyperbola

Eccentricity of a Hyperbola

- The hyperbola can be wide or narrow

Eccentricity of a Hyperbola

- As with eccentricity of an hyperbola, the formula is
- Note that for hyperbolas c > a
- Thus eccentricity > 1

Try It Out

- If the vertices are (1, 6) and (1, 8) and the eccentricity is 5/2
- Find the equation (standard form) of the hyperbola

- The center of the hyperbola is at (-3, -3) and the conjugate axis has length 6, and the eccentricity = 2
- Find two possible hyperbola equations

Application – Locating Position

- For any point on a hyperbolic curve
- Difference between distances to foci is constant.

- Result: hyperbolas can be used to locate enemy guns

“If the sound of an enemy gun is heard at two listening posts and the difference in time is calculated, then the gun is known to be located on a particular hyperbola. A third listening post will determine a second hyperbola, and then the gun emplacement can be spotted as the intersection of the two hyperbolas.”

Application – Locating Position

- The loran system navigator
- equipped with a map that gives curves, called loran lines of position.
- Navigators find the time interval between these curves,
- Narrow down the area that their craft's position is in.

- Then switch to a different pair of loran transmitters
- Repeat the procedure
- Find another curve representing the craft's position.

Construction

- Consider a the blue string
- Keep markeragainst rulerand with stringtight
- Keep end ofruler on focusF1 , string tied to other end

Graphing a Hyperbola on the TI

- As with the ellipse, the hyperbola is not a function
- Possible to solve for y
- Get two expressions
- Graph each

- What happensif it opensright and left?

Graphing a Hyperbola on the TI

- Top and bottom of hyperbola branches are graphed separately
- As with ellipsesyou must

Ellipse with Geogebra

Assignment

- Hyperbolas 2
- Exercise Set 6.3
- Exercises 27 – 31 oddand 49 – 63 odd

Download Presentation

Connecting to Server..