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.
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:
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: .
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:
This means that all the things we know how to calculate also change:
, , ,
This means that for a hyperbola :
, , ,
This means that is valid equation.
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
If we insert instead of y into the other equation we’ll get: