Hyperbola Activity: Extending w/ Foci, Assymptotes & Point A

In summary, the conversation discusses extending an activity on conic sections, specifically hyperbolas. The speaker asks for help in finding the foci, asymptotes, and "point a" of a hyperbola. They also mention that they have included attachments since they do not have a camera or scanner. The respondent explains that to find these points, one must have three points on the hyperbola and make measurements. They also mention that constructing the foci is more complicated and involves following their definitions. The conversation also briefly touches on the number of parameters and degrees of freedom in a normalised quadratic equation and how it relates to hyperbolas.
  • #1
thimkepeng
1
0
I need to extend this activity somehow, but I forgot this stuff already? I learned this a long time ago, I think this activity is too simple so can someone tell me how to find the foci, assymptotes, etc, and what the "point a" is for?
http://mste.illinois.edu/courses/ci399TSMsu03/folders/jmpeter1/Daily%20Assignments/Conic%20Sections/Hyperbola%20Paper%20Folding%28CI399%29.html
I put all I can do in the attachments since I don't have a camera or scanner:
 

Attachments

  • hyperbola.png
    hyperbola.png
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  • #2
Hyperbola is one of a class of functions known as "conic sections". Use google to find out about them.
Once you have three points on the hyperbola, you can make some measurements to determine the whole thing.
Constructing the foci etc is a bit trickier - you have to follow their definitions.
 
  • #3
Simon Bridge said:
Once you have three points on the hyperbola, you can make some measurements to determine the whole thing.
It's three for a circle, four for a parabola, five for ellipse or hyperbola.
A normalised quadratic equation in two variables has five parameters. The classifications ellipse and hyperbola set constraints on the ranges of the parameters, but no exact relationships, so five degrees of freedom.
If you regard two hyperbolae as the same if they can rotated and translated to line up then there are only two degrees of freedom.
 
  • #4
Sorry, I wasn't clear.
This method of construction specifies the foci at the start.
 

What is a hyperbola?

A hyperbola is a type of conic section that is formed when a plane intersects with a cone at an angle. It is a curved shape with two symmetrical halves that are mirrored across a center line called the transverse axis.

What are the foci and how do they relate to a hyperbola?

The foci are two fixed points inside a hyperbola that are used to define its shape and size. The distance between the foci, called the major axis, determines the overall width of the hyperbola. The foci also play a role in determining the asymptotes of a hyperbola.

What are asymptotes and how are they related to a hyperbola?

Asymptotes are imaginary lines that a hyperbola approaches but never crosses. They are formed by extending the sides of the rectangle formed by the transverse and conjugate axes. Asymptotes help define the shape of a hyperbola and can be used to graph it accurately.

How do you calculate the coordinates of Point A on a hyperbola?

Point A, also known as the vertex, is the point where the hyperbola crosses the transverse axis. To find its coordinates, you can use the equation (h, k), where h is the x-coordinate and k is the y-coordinate. The values of h and k can be calculated using the foci and the distance between them.

What are some real-life applications of hyperbolas?

Hyperbolas have many real-life applications, such as in satellite orbits, radio wave transmissions, and the design of curved mirrors and lenses. They are also used in economics to represent supply and demand curves, and in physics to describe the path of a projectile under the influence of gravity.

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