How Do You Calculate the Axes of a Hyperbola Given Its Foci and a Point on It?

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Homework Help Overview

The discussion revolves around calculating the axes of a hyperbola given its foci and a point on it. The problem involves understanding the definitions and properties of hyperbolas, particularly focusing on the relationship between the foci and the axes.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the definition of a hyperbola and its equation, questioning how to derive the values of the semi-major and semi-minor axes from given points. There are attempts to relate the distances from the foci to the axes, and some participants express uncertainty about the next steps in calculations.

Discussion Status

The discussion is active with various interpretations of the hyperbola's properties being explored. Some participants have provided guidance on using the relationship between the foci and the axes, while others are questioning the specifics of the hyperbola's equation and its parameters.

Contextual Notes

There is a mention of specific values and relationships, such as the eccentricity and the distances involved, but the exact values and further calculations remain unresolved. Participants are also discussing the implications of different forms of the hyperbola's equation.

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Homework Statement


Points E and F are the focuses of the hyperbola and point X are on the hyperbola. Determine the size of the main and minor half-axes of the hyperbola.

upload_2019-3-20_11-19-54.png


Homework Equations


x2 = e2 - f2
x = 8

The Attempt at a Solution


I think that eccentricity is 4 units (x/2). But I don’t know how to continue.
 

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How is the hyperbola defined? What is the equation of the hyperbola?
 
A hyperbola is a set of points, such that for any point
b4dc73bf40314945ff376bd363916a738548d40a
of the set, the absolute difference of the distances to two fixed points F1, F2 is constant = 2a

equation...f(x)=1/x
 
charlie05 said:
A hyperbola is a set of points, such that for any point
b4dc73bf40314945ff376bd363916a738548d40a
of the set, the absolute difference of the distances to two fixed points F1, F2 is constant =
Correct, What is the name for a?
What is the value of a of the hyperbola shown in the problem?
charlie05 said:
equation...f(x)=1/x
That is one special hyperbola. Is it the same as the one in the problem?
 
a is semi major axis
use PF2-PF1=2a?
XF-EX=2a...10-6=2a...a=2cm ?
 
charlie05 said:
a is semi major axis
use PF2-PF1=2a?
XF-EX=2a...10-6=2a...a=2cm ?
Correct!
Where do the semimajor and minor axes appear in the equation of the hyperbola?
Can you draw a hyperbola, with axes parallel with the coordinate axes? What is the equation of such hyperbola?
 
[x^2/a^2] - [y^1/b^2] = 1

a - semimajor axe
b - minor axe
 
Last edited:
charlie05 said:
[x^2/a^2] - [y^1/b^2] = 1

a - semimajor axe
b - minor axe
It should be x^2/a^2 - y^2/b^2 = 1
You know a, and the coordinates of the point X : Substitute into the equation of the hyperbola. Calculate b.
 

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