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

[charlie05]

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.

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.

 

[ehild]

How is the hyperbola defined? What is the equation of the hyperbola?

 

[charlie05]

A hyperbola is a set of points, such that for any point

of the set, the absolute difference of the distances to two fixed points F1, F2 is constant = 2a

equation...f(x)=1/x

 

[ehild]

A hyperbola is a set of points, such that for any point

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?

equation...f(x)=1/x

That is one special hyperbola. Is it the same as the one in the problem?

 

[charlie05]

a is semi major axis
use PF2-PF1=2a?
XF-EX=2a...10-6=2a...a=2cm ?

 

[ehild]

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?

 

[charlie05]

[x^2/a^2] - [y^1/b^2] = 1

a - semimajor axe
b - minor axe

 

[ehild]

[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.