# Triangle and hyperbola

• charlie05
In summary, the main and minor half-axes of the hyperbola can be determined by using the equation x^2 = e^2 - f^2 and the given coordinates of point X. The eccentricity is 4 units and the equation of the hyperbola is f(x) = 1/x. The semi major axis is a=2cm and the semi minor axis is b=1cm. The equation of the hyperbola is x^2/4 - y^2 = 1.

## 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.

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.

#### Attachments

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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
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
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.

## 1. What is the difference between a triangle and a hyperbola?

A triangle is a two-dimensional shape with three sides and three angles, while a hyperbola is a type of curve that is created when a plane intersects with a cone at an angle. Triangles have a finite area, while hyperbolas extend infinitely in both directions.

## 2. How many types of triangles and hyperbolas are there?

There are three types of triangles: equilateral, isosceles, and scalene. There are also three types of hyperbolas: horizontal, vertical, and oblique. In addition, there are different classifications of hyperbolas based on their eccentricity, such as elliptical, parabolic, and hyperbolic.

## 3. What are the properties of a triangle and a hyperbola?

The properties of a triangle include having three sides, three angles that add up to 180 degrees, and the sum of any two sides must be greater than the third side. The properties of a hyperbola include having two symmetrical curves that are mirror images of each other, and the distance between any point on the curve and the two foci is constant.

## 4. What are the real-world applications of triangles and hyperbolas?

Triangles have many real-world applications, such as in architecture, engineering, and navigation. They are also used in trigonometry to calculate distances and angles. Hyperbolas are used in physics, astronomy, and engineering to describe the paths of objects in motion, such as comets and satellites.

## 5. How are triangles and hyperbolas related to each other?

Triangles and hyperbolas are related through the Pythagorean theorem. In a right triangle, the length of the hypotenuse can be calculated using the lengths of the other two sides. This relationship is also seen in the equation of a hyperbola, where the distance between any point on the curve and the two foci is equal to the difference between the lengths of the two axes.