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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 F_1 and F_2 are solid points of a plain M and a a positive real number such that a < \frac{1}{2} \mid F_1 F_2 \mid.  Hyperbola is a set of all points of a plane M for which the absolute value of the differences from points F_1 and F_2 is a constant and equal to 2a, 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 B.

b point on hyperbola

 

\mid F_2 B \mid - \mid B F_1 \mid = 2 \cdot 3 = 6

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

\{ T \in \mathbb{M} : \mid \mid T F_1 \mid - \mid T F_2 \mid \mid = 2a \}


How to draw a hyperbola?

First, you need to find the foci (F_1, F_2), centre (point O) and vertexes (A, B).

Centre of the hyperbola is the middle point of a line \box{F_1 F_2}.

Line \box{AB} is called the real axis and lines \box{OA} and \box{OB} are called real semi-axis.

real axis and semi-real axis

Number e = \frac{1}{2} \mid F_1 F_2 \mid is called linear eccentricity of a hyperbola, and b = \sqrt{e^2 - a^2} is the length of an imaginary axis of a hyperbola.

elements of hyperbola

Number \varepsilon = \frac{e}{a} 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: y_1 = - \frac{b}{a}x and y_2 = \frac{b}{a}x.


 

How to constructing a hyperbola?

Let’s say that we are given foci F_1, F_2 and the length of real axis 2a. We want to construct few points of a hyperbola from these information.

First, we’ll construct the center O, as a middle point of a line \box{F_1 F_2}.

Somewhere on the side we’ll draw a line \overline{UV} whose length is equal to 2a and then extend it over V.

constructing hyperbola

On that extension we’ll choose a point P_1 and construct circles c_1 (F_1, \mid UP_1 \mid), c_2(F_2, \mid UP_1 \mid). The intersections of these circles will give us four different points of a hyperbola.

constructing hyperbola 2

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

constructing hyperbola 3


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

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Where:

b^2 = e^2 - a^2.

From this formula you’ll know the foci coordinates, F_1 (- e, 0), F_2 (e, 0) and the vertexes A(- a, 0), B ( a, 0).

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 y_1 = - \frac{b}{a}x and y_2 = \frac{b}{a}x you can directly find their equations.

Example 1. Draw a hyperbola with equation \frac{x^2}{16} - \frac{y^2}{9} = 1.

a = \sqrt{16} = 4

b = \sqrt{9} = 3

e^2 = a^2 + b^2 = 16 + 9 = 25 \rightarrow e = 5

F_1= (-5, 0), F_2= (5, 0),  A= (-4, 0),  B= (4, 0)

First asymptote: y_1 = -\frac{3}{4}x.

Second asymptote: y_2 = \frac{3}{4}x.

 


What if the x and y variables changed their places?

Let’s examine a hyperbola given with an equation \frac{y^2}{16} - \frac{x^2}{9} = 1

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

Equation of that hyperbola is:

\frac{x^2}{b^2} - \frac{y^2}{a^2} = - 1

Or:

\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1.

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

F_1 = (0, - e)F_2 = (e, 0)A (0, - a)B (0, a)

This means that for a hyperbola \frac{y^2}{16} - \frac{x^2}{9} = 1 :

F_1 = (0, - 5)F_2 = (5, 0)A (0, - 4)B (0, 4)


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


Isosceles hyperbola

This means that a = b is valid equation.

E... \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Or:

x^2 - y^2 = a^2.

Example 2. Draw a hyperbola x^2 - y^2 = 16

a = 4,   y_1 = x,   y_2 = - x

The condition to hyperbola and a line to meet

If we want to algebraically determine an intersection of a line l... k = kx + l, which is not an asymptote of a hyperbola, and a hyperbola H... \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 If we insert kx + linstead of y into the other equation we'll get: \frac{x^2}{a^2} – \frac{(kx + l)^2}{b^2} = 1$

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