Definition. A hyperbola is the locus of points on the plane y; the absolute value of the difference in the distances of each of them from two given points of this plane, called the foci of y, is a constant value, provided that this value is not zero and is less than the distance between the foci.
Let us denote the distance between the foci by a constant value equal to the modulus of the difference in distances from each point of the hyperbola to the foci, by (by condition ). As in the case of an ellipse, we draw the abscissa axis through the foci, and take the middle of the segment as the origin of coordinates (see Fig. 44). The foci in such a system will have coordinates. We derive the equation of the hyperbola in the selected coordinate system. By the definition of a hyperbola, for any point of it we have or
But . Therefore we get
After simplifications similar to those made when deriving the equation of the ellipse, we obtain the following equation:
which is a consequence of equation (33).
It is easy to see that this equation coincides with equation (27) obtained for an ellipse. However, in equation (34) the difference is , since for a hyperbola . Therefore we put
Then equation (34) is reduced to the following form:
This equation is called the canonical hyperbola equation. Equation (36), as a consequence of equation (33), is satisfied by the coordinates of any point of the hyperbola. It can be shown that the coordinates of points that do not lie on the hyperbola do not satisfy equation (36).
Let us establish the form of the hyperbola using its canonical equation. This equation contains only even powers of the current coordinates. Consequently, a hyperbola has two axes of symmetry, in this case coinciding with the coordinate axes. In what follows, we will call the axes of symmetry of a hyperbola the axes of the hyperbola, and the point of their intersection the center of the hyperbola. The axis of the hyperbola on which the foci are located is called the focal axis. Let us examine the form of the hyperbola in the first quarter, where
Here, since otherwise y would take imaginary values. As x increases from a to, it increases from 0 to. Part of the hyperbola lying in the first quarter will be the arc shown in Fig. 47.
Since the hyperbola is located symmetrically relative to the coordinate axes, this curve has the form shown in Fig. 47.
The intersection points of a hyperbola with the focal axis are called its vertices. Assuming hyperbolas in the equation, we find the abscissas of its vertices: . Thus, a hyperbola has two vertices: . The hyperbola does not intersect with the ordinate axis. In fact, by putting hyperbolas in the equation we obtain imaginary values for y: . Therefore, the focal axis of a hyperbola is called the real axis, and the axis of symmetry perpendicular to the focal axis is called the imaginary axis of the hyperbola.
The real axis is also called a segment connecting the vertices of a hyperbola, and its length is 2a. The segment connecting the points (see Fig. 47), as well as its length, is called the imaginary axis of the hyperbola. The numbers a and b are respectively called the real and imaginary semi-axes of the hyperbola.
Let us now consider a hyperbola located in the first quarter and which is the graph of the function
Let us show that the points of this graph, located at a sufficiently large distance from the origin of coordinates, are arbitrarily close to the straight line
passing through the origin and having a slope
For this purpose, consider two points having the same abscissa and lying respectively on the curve (37) and straight line (38) (Fig. 48), and make up the difference between the ordinates of these points
The numerator of this fraction is a constant value, and the denominator increases indefinitely with unlimited increase. Therefore, the difference tends to zero, i.e. points M and N come closer together indefinitely as the abscissa increases indefinitely.
From the symmetry of the hyperbola with respect to the coordinate axes it follows that there is one more straight line to which the points of the hyperbola are arbitrarily close at an unlimited distance from the origin. Direct
are called asymptotes of the hyperbola.
In Fig. 49 shows the relative position of the hyperbola and its asymptotes. This figure also shows how to construct the asymptotes of a hyperbola.
To do this, construct a rectangle with a center at the origin and with sides parallel to the axes and correspondingly equal to . This rectangle is called the main rectangle. Each of its diagonals, extended indefinitely in both directions, is an asymptote of a hyperbola. Before constructing a hyperbola, it is recommended to construct its asymptotes.
The ratio of half the distance between the foci to the real semi-axis of the hyperbola is called the eccentricity of the hyperbola and is usually denoted by the letter:
Since for a hyperbola, the eccentricity of the hyperbola is greater than one: Eccentricity characterizes the shape of the hyperbola
Indeed, from formula (35) it follows that . From this it is clear that the smaller the eccentricity of the hyperbola,
the smaller the ratio of its semi-axes. But the relation determines the shape of the main rectangle of the hyperbola, and therefore the shape of the hyperbola itself. The lower the eccentricity of the hyperbola, the more elongated its main rectangle is (in the direction of the focal axis).
Hyperbola and parabola
Let's move on to the second part of the article about second order lines, dedicated to two other common curves - hyperbole And parabola. If you came to this page from a search engine or have not yet had time to navigate the topic, then I recommend that you first study the first section of the lesson, in which we examined not only the main theoretical points, but also got acquainted with ellipse. I suggest that the rest of the readers significantly expand their school knowledge about parabolas and hyperbolas. Hyperbola and parabola - are they simple? ...Can't wait =)
Hyperbola and its canonical equation
The general structure of the presentation of the material will resemble the previous paragraph. Let's start with the general concept of a hyperbola and the task of constructing it.
The canonical equation of a hyperbola has the form , where are positive real numbers. Please note that, unlike ellipse, the condition is not imposed here, that is, the value of “a” may be less than the value of “be”.
I must say, quite unexpectedly... the equation of the “school” hyperbola does not even closely resemble the canonical notation. But this mystery will still have to wait for us, but for now let’s scratch our heads and remember what characteristic features the curve in question has? Let's spread it on the screen of our imagination graph of a function ….
A hyperbola has two symmetrical branches.
A hyperbole has two asymptotes.
Not bad progress! Any hyperbole has these properties, and now we will look with genuine admiration at the neckline of this line:
Example 4
Construct the hyperbola given by the equation
Solution: in the first step, we bring this equation to canonical form. Please remember the standard procedure. On the right you need to get “one”, so we divide both sides of the original equation by 20: 
Here you can reduce both fractions, but it is more optimal to do each of them three-story:
And only after that carry out the reduction:
Select the squares in the denominators:
Why is it better to carry out transformations this way? After all, the fractions on the left side can be immediately reduced and obtained. The fact is that in the example under consideration we were a little lucky: the number 20 is divisible by both 4 and 5. In the general case, such a number does not work. Consider, for example, the equation . Here everything is sadder with divisibility and without three-story fractions no longer possible: 
So, let's use the fruit of our labors - the canonical equation:
How to construct a hyperbola?
There are two approaches to constructing a hyperbola - geometric and algebraic.
From a practical point of view, drawing with a compass... I would even say utopian, so it is much more profitable to once again use simple calculations to help.
It is advisable to adhere to the following algorithm, first the finished drawing, then the comments:
1) First of all, we find asymptotes. If a hyperbola is given by a canonical equation, then its asymptotes are straight 
. In our case: 
. This item is required! This is a fundamental feature of the drawing, and it will be a blunder if the branches of the hyperbola “crawl out” beyond their asymptotes.
2) Now we find two vertices of a hyperbola, which are located on the abscissa axis at points 
. The derivation is elementary: if , then the canonical equation turns into , from which it follows that . The hyperbola under consideration has vertices 
3) We are looking for additional points. Usually 2-3 are enough. In the canonical position, the hyperbola is symmetrical with respect to the origin and both coordinate axes, so it is enough to carry out the calculations for the 1st coordinate quarter. The technique is exactly the same as when constructing ellipse. From the canonical equation in the draft we express:
The equation breaks down into two functions: 
– determines the upper arcs of the hyperbola (what we need); 
– defines the lower arcs of a hyperbola.
This suggests finding points with abscissas: 
4) Let us depict the asymptotes in the drawing , peaks , additional and symmetrical points to them in other coordinate quarters. Carefully connect the corresponding points at each branch of the hyperbola:
Technical difficulty may arise with irrational slope, but this is a completely surmountable problem.
Line segment called real axis hyperboles,
its length is the distance between the vertices;
number 
called real semi-axis hyperbole;
number – imaginary semi-axis.
In our example: 
, and, obviously, if this hyperbola is rotated around the center of symmetry and/or moved, then these values won't change.
Definition of hyperbole. Foci and eccentricity
The hyperbole has the same thing as the ellipse, there are two special points called tricks. I didn’t say anything, but just in case someone misunderstands: the center of symmetry and focal points, of course, do not belong to curves.
The general concept of the definition is also similar:
Hyperbole called the set of all points in the plane, absolute value the difference in distances to each of which from two given points is a constant value, numerically equal to the distance between the vertices of this hyperbola: . In this case, the distance between the foci exceeds the length of the real axis: .
If a hyperbola is given by a canonical equation, then distance from the center of symmetry to each focus calculated using the formula: .
And, accordingly, the foci have coordinates 
.
For the hyperbola under study: 
Let's understand the definition. Let us denote by the distances from the foci to an arbitrary point of the hyperbola: 
First, mentally move the blue dot along the right branch of the hyperbola - wherever we are, module(absolute value) of the difference between the lengths of the segments will be the same:
If you “throw” the point onto the left branch and move it there, then this value will remain unchanged.
The modulus sign is needed because the difference in lengths can be either positive or negative. By the way, for any point on the right branch 
(since the segment is shorter than the segment ). For any point on the left branch the situation is exactly the opposite and 
.
Moreover, in view of the obvious property of the module, it does not matter what is subtracted from what.
Let's make sure that in our example the module of this difference is really equal to the distance between the vertices. Mentally place the point at the right vertex of the hyperbola. Then: , which is what needed to be checked.
A hyperbola is a set of points on a plane, the difference in distances from two given points, foci, is a constant value and equal to .
Similarly to the ellipse, we place the foci at points , (see Fig. 1).
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It is known that in a triangle the difference between two sides is less than the third side, so, for example, with we get:
Let’s bring both sides to the square and after further transformations we find:
Where . The hyperbola equation (1) is canonical hyperbola equation.
The hyperbola is symmetrical with respect to the coordinate axes, therefore, as for the ellipse, it is enough to plot its graph in the first quarter, where:
Range of values for the first quarter.
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Form and characteristics of a hyperbola
Let us examine equation (1) the shape and location of the hyperbola.
- Variables and are included in equation (1) in pair powers. Therefore, if a point belongs to a hyperbola, then the points also belong to a hyperbola. This means that the figure is symmetrical about the axes and and the point, which is called the center of the hyperbola.
- Let's find the points of intersection with the coordinate axes. Substituting into equation (1) we find that the hyperbola intersects the axis at points . Putting it, we get an equation that has no solutions. This means that the hyperbola does not intersect the axis. The points are called vertices of the hyperbola. The segment = and is called the real axis of the hyperbola, and the segment is called the imaginary axis of the hyperbola. The numbers and are called the real and imaginary semi-axes of the hyperbola, respectively. The rectangle created by the axes is called the principal rectangle of the hyperbola.
- From equation (1) it turns out that , that is . This means that all points of the hyperbola are located to the right of the line (right branch of the hyperbola) and to the left of the line (left branch of the hyperbola).
- Let's take a point on the hyperbola in the first quarter, that is, and therefore . Since 0" title="Rendered by QuickLaTeX.com" height="31" width="156" style="vertical-align: -12px;"> 0" title="Rendered by QuickLaTeX.com" height="31" width="156" style="vertical-align: -12px;"> , при title="Rendered by QuickLaTeX.com" height="12" width="51" style="vertical-align: 0px;"> title="Rendered by QuickLaTeX.com" height="12" width="51" style="vertical-align: 0px;"> , тогда функция монотонно возрастает при title="Rendered by QuickLaTeX.com" height="12" width="51" style="vertical-align: 0px;"> title="Rendered by QuickLaTeX.com" height="12" width="51" style="vertical-align: 0px;"> . Аналогично, так как при title="Rendered by QuickLaTeX.com" height="12" width="51" style="vertical-align: 0px;"> title="Rendered by QuickLaTeX.com" height="12" width="51" style="vertical-align: 0px;"> , тогда функция выпуклая вверх при title="Rendered by QuickLaTeX.com" height="12" width="51" style="vertical-align: 0px;"> title="Rendered by QuickLaTeX.com" height="12" width="51" style="vertical-align: 0px;"> .!}
Asymptotes of a hyperbola
There are two asymptotes of a hyperbola. Let's find the asymptote to the branch of the hyperbola in the first quarter, and then use the symmetry. Consider the point in the first quarter, that is. In this case, , then the asymptote has the form: , where
This means that the straight line is the asymptote of the function. Therefore, due to symmetry, the asymptotes of a hyperbola are straight lines.
Using the established characteristics, we will construct a branch of the hyperbola, which is located in the first quarter, and use the symmetry:
Rice. 2
In the case when , that is, the hyperbola is described by the equation. This hyperbola contains asymptotes, which are the bisectors of the coordinate angles.
Examples of problems on constructing a hyperbola
Example 1
Task
Find the axes, vertices, foci, eccentricity and equations of asymptotes of the hyperbola. Construct a hyperbola and its asymptotes.
Solution
Let's reduce the hyperbola equation to canonical form:
Comparing this equation with the canonical (1) we find , , . Peaks, focuses and . Eccentricity; asptotes; We are building a parabola. (see Fig. 3)
Write the equation of the hyperbola:
Solution
By writing the asymptote equation in the form we find the ratio of the semi-axes of the hyperbola. According to the conditions of the problem, it follows that . Therefore, the Problem was reduced to solving a system of equations:
Substituting into the second equation of the system, we get:
where . Now we find.
Therefore, the hyperbola has the following equation:
Answer
.Hyperbola and its canonical equation updated: June 17, 2017 by: Scientific Articles.Ru
Hello, dear Argemona University students! Welcome to another lecture on the magic of functions and integrals.
Today we will talk about hyperbole. Let's start simple. The simplest type of hyperbole is:
This function, unlike the straight line in its standard forms, has a special feature. As we know, the denominator of a fraction cannot be zero, because you cannot divide by zero.
x ≠ 0
From here we conclude that the domain of definition is the entire number line, except for point 0: (-∞; 0) ∪ (0; +∞).
If x tends to 0 from the right (written like this: x->0+), i.e. becomes very, very small, but remains positive, then y becomes very, very large positive (y->+∞).
If x tends to 0 from the left (x->0-), i.e. becomes very, very small in absolute value, but remains negative, then y will also be negative, but in absolute value it will be very large (y->-∞).
If x tends to plus infinity (x->+∞), i.e. becomes a very large positive number, then y will become an increasingly smaller positive number, i.e. will tend to 0, remaining positive all the time (y->0+).
If x tends to minus infinity (x->-∞), i.e. becomes large in modulus, but a negative number, then y will also always be a negative number, but small in modulus (y->0-).
Y, like x, cannot take the value 0. It only tends to zero. Therefore, the set of values is the same as the domain of definition: (-∞; 0) ∪ (0; +∞).
Based on these considerations, we can schematically draw a graph of the function
It can be seen that the hyperbola consists of two parts: one is located in the 1st coordinate angle, where the x and y values are positive, and the second part is in the third coordinate angle, where the x and y values are negative.
If we move from -∞ to +∞, then we see that our function decreases from 0 to -∞, then there is a sharp jump (from -∞ to +∞) and the second branch of the function begins, which also decreases, but from +∞ to 0. That is, this hyperbole is decreasing.
If you change the function just a little: use the magic of minus,
(1")
Then the function will miraculously move from the 1st and 3rd coordinate quarters to the 2nd and 4th quarters and become increasing.
Let me remind you that the function is increasing, if for two values x 1 and x 2 such that x 1<х 2 , значения функции находятся в том же отношении f(х 1) < f(х 2).
And the function will be decreasing, if f(x 1) > f(x 2) for the same values of x.
The branches of the hyperbola approach the axes, but never intersect them. Lines that the graph of a function approaches but never intersects are called asymptote this function.
For our function (1), the asymptotes are the straight lines x=0 (OY axis, vertical asymptote) and y=0 (OX axis, horizontal asymptote).
Now let's complicate the simplest hyperbola a little and see what happens to the graph of the function.
(2)
We just added the constant “a” to the denominator. Adding a number to the denominator as a term to x means moving the entire “hyperbolic construction” (along with the vertical asymptote) (-a) positions to the right if a is a negative number, and (-a) positions to the left if a is a positive number.
On the left graph, a negative constant is added to x (a<0, значит, -a>0), which causes the graph to move to the right, and on the right graph there is a positive constant (a>0), due to which the graph is moved to the left.
And what magic can affect the transfer of the “hyperbolic structure” up or down? Adding a constant term to a fraction.
(3)
Now our entire function (both branches and the horizontal asymptote) will rise b positions up if b is a positive number, and will go down b positions if b is a negative number.
Please note that the asymptotes move along with the hyperbola, i.e. the hyperbola (both of its branches) and both of its asymptotes must necessarily be considered as an inseparable structure that moves uniformly to the left, right, up or down. It’s a very pleasant feeling when by just adding a number you can make the entire function move in any direction. What is not magic, which you can master very easily and direct it at your discretion in the right direction?
By the way, you can control the movement of any function this way. In the next lessons we will consolidate this skill.
Before assigning you homework, I would like to draw your attention to this function:
(4)
The lower branch of the hyperbola moves from the 3rd coordinate angle upward - to the second, to the angle where the value of y is positive, i.e. this branch is reflected symmetrically relative to the OX axis. And now we get an even function.
What does "even function" mean? The function is called even, if the condition is met: f(-x)=f(x)
The function is called odd, if the condition is met: f(-x)=-f(x)
In our case
(5)
Every even function is symmetrical about the OY axis, i.e. a parchment with a drawing of a graph can be folded along the OY axis, and the two parts of the graph will exactly coincide with each other.
As we can see, this function also has two asymptotes - horizontal and vertical. Unlike the functions discussed above, this function is increasing on one part and decreasing on the other.
Let's now try to manipulate this graph by adding constants.
(6)
Recall that adding a constant as a term to “x” causes the entire graph (along with the vertical asymptote) to move horizontally, along the horizontal asymptote (to the left or to the right, depending on the sign of this constant).
(7)
And adding the constant b as a term to a fraction causes the graph to move up or down. Everything is very simple!
Now try experimenting with this magic yourself.
Homework 1.
Everyone takes two functions for their experiments: (3) and (7).
a=the first digit of your LD
b=second digit of your LD
Try to get to the magic of these functions, starting with the simplest hyperbola, as I did in the lesson, and gradually adding your own constants. You can already model function (7) based on the final form of function (3). Indicate the domains of definition, the set of values, and asymptotes. How functions behave: decrease, increase. Even odd. In general, try to do the same research as you did in class. Perhaps you will find something else that I forgot to talk about.
By the way, both branches of the simplest hyperbola (1) are symmetrical with respect to the bisector of the 2nd and 4th coordinate angles. Now imagine that the hyperbola began to rotate around this axis. Let's get such a nice figure that can be used.
Task 2. Where can this figure be used? Try to draw a rotation figure for function (4) relative to its axis of symmetry and think about where such a figure could find application.
Remember how at the end of the last lesson we got a straight line with a punctured point? And here's the last one task 3.
Construct a graph of this function:
(8)
Coefficients a, b are the same as in task 1.
c=third digit of your LD or a-b if your LD is two-digit.
A little hint: first, the fraction obtained after substituting the numbers must be simplified, and then you will get an ordinary hyperbola, which you need to construct, but in the end you must take into account the domain of definition of the original expression.
