# Hyperbola

Hyperbola is not a closed curve, but its parts are in a way bounded by two lines – asymptotes. Do you know that some astronomical objects and specific protons are moving in a way we can’t explain using only circles and ellipses? For these movements it is important to introduce another second degree curve.

It is uniquely defined by its foci and a vertex.

Suppose points and  are solid points of a plain and a a positive real number such that .  Hyperbola is a set of all points of a plane for which the absolute value of the differences from points  and  is a constant and equal to , where a is the distance of a vertex to the center of a hyperbola.

This definition may seem a little bit complicated so we’ll explain it using a drawing.

Let’s choose any point .

This equality will be valid for all points on a hyperbola. In other words, we can define it as follows:

## How to draw a hyperbola?

First, you need to find the foci , centre (point ) and vertexes (, ).

Centre of the hyperbola is the middle point of a line .

Line is called the real axis and lines  and  are called real semi-axis.

Number is called linear eccentricity of a hyperbola, and is the length of an imaginary axis of a hyperbola.

Number is called the numeric eccentricity of a hyperbola.

The last step to do is to draw the asymptotes. There will be two asymptotes with equations: and .

## How to constructing a hyperbola?

Let’s say that we are given foci and the length of real axis . We want to construct few points of a hyperbola from these information.

First, we’ll construct the center , as a middle point of a line .

Somewhere on the side we’ll draw a line whose length is equal to and then extend it over .

On that extension we’ll choose a point and construct circles . The intersections of these circles will give us four different points of a hyperbola.

For more points you’ll use more other points on that extension, draw the circles and mark their intersections as new points.

## Hyperbola equation

Just like in an ellipse, here we’ll also examine a hyperbola in a special position relating the coordinate system. We’ll set it in a way that the center of a hyperbola falls into the origin.

Suppose is a hyperbola whose foci lie on the – axis and its centre is the origin of a coordinate system. Then, the hyperbola is given with an equation:

Where:

.

From this formula you’ll know the foci coordinates, , and the vertexes , .

Now you know almost everything you need to know in order to construct a hyperbola from its equation properly.

We already mentioned another important thing – asymptotes. Since and  you can directly find their equations.

Example 1. Draw a hyperbola with equation .

, ,  ,

First asymptote: .

Second asymptote: .

What if the and variables changed their places?

Let’s examine a hyperbola given with an equation

Now, the foci will be on the y – axis, and not on the x-axis. In other words, x – axis and y – axis changed their roles.

Suppose is a hyperbola whose foci lie on – axis and its center is the origin of the coordinate system.

Equation of that hyperbola is:

Or:

.

This means that all the things we know how to calculate also change:

This means that for a hyperbola :

A hyperbola whose real and imaginary axis are the same is called the isosceles hyperbola.

## Isosceles hyperbola

This means that is valid equation.

Or:

.

Example 2. Draw a hyperbola

,   ,

The condition to hyperbola and a line to meet

If we want to algebraically determine an intersection of a line , which is not an asymptote of a hyperbola, and a hyperbola \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 y = kx + l\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 kx + l\frac{x^2}{a^2} – \frac{(kx + l)^2}{b^2} = 1\$

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