General view of a hyperbole. Definition of hyperbole. Foci and eccentricity

Definition. A hyperbola is the locus of points in the plane absolute value the difference in the distances of each of them from two given points of this plane, called foci, y has 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 chosen 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 a 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).

A hyperbola is the locus of points on a plane, the modulus of the difference in distances from each of them to two given points F_1 and F_2 is a constant value (2a), less than the distance (2c) between these given points (Fig. 3.40, a). This geometric definition expresses focal property of a hyperbola.

Focal property of a hyperbola

Points F_1 and F_2 are called the foci of the hyperbola, the distance 2c=F_1F_2 between them is the focal length, the middle O of the segment F_1F_2 is the center of the hyperbola, the number 2a is the length of the real axis of the hyperbola (accordingly, a is the real semi-axis of the hyperbola). The segments F_1M and F_2M connecting an arbitrary point M of the hyperbola with its foci are called focal radii of the point M. The segment connecting two points of a hyperbola is called a chord of the hyperbola.

The relation e=\frac(c)(a) , where c=\sqrt(a^2+b^2) , is called eccentricity of the hyperbola. From definition (2a<2c) следует, что e>1 .

Geometric definition of hyperbola, expressing its focal property, is equivalent to its analytical definition - the line given by the canonical hyperbola equation:

\frac(x^2)(a^2)-\frac(y^2)(b^2)=1.

Indeed, let us introduce a rectangular coordinate system (Fig. 3.40, b). We take the center O of the hyperbola as the origin of the coordinate system; We will take the straight line passing through the foci (focal axis) as the abscissa axis (the positive direction on it is from point F_1 to point F_2); Let us take a straight line perpendicular to the abscissa axis and passing through the center of the hyperbola as the ordinate axis (the direction on the ordinate axis is chosen so that the rectangular coordinate system Oxy is right).

Let's create an equation for a hyperbola using a geometric definition expressing the focal property. In the selected coordinate system, we determine the coordinates of the foci F_1(-c,0) and F_2(c,0) . For an arbitrary point M(x,y) belonging to a hyperbola, we have:

\left||\overrightarrow(F_1M)|-|\overrightarrow(F_2M)|\right|=2a.

Writing this equation in coordinate form, we get:

\sqrt((x+c)^2+y^2)-\sqrt((x-c)^2+y^2)=\pm2a.

Performing transformations similar to those used in deriving the ellipse equation (i.e., getting rid of irrationality), we arrive at the canonical hyperbola equation:

\frac(x^2)(a^2)-\frac(y^2)(b^2)=1\,

where b=\sqrt(c^2-a^2) , i.e. the chosen coordinate system is canonical.

Carrying out the reasoning in reverse order, we can show that all points whose coordinates satisfy equation (3.50), and only they, belong to the locus of points called a hyperbola. Thus the analytical definition of a hyperbola is equivalent to its geometric definition.

Directorial property of a hyperbola

The directrixes of a hyperbola are two straight lines passing parallel to the ordinate axis of the canonical coordinate system at the same distance a^2\!\!\not(\phantom(|))\,c from it (Fig. 3.41, a). When a=0, when the hyperbola degenerates into a pair of intersecting lines, the directrixes coincide.

A hyperbola with eccentricity e=1 can be defined as the locus of points in the plane, for each of which the ratio of the distance to a given point F (focus) to the distance to a given straight line d (directrix) not passing through given point, constant and equal to eccentricity e ( directorial property of a hyperbola). Here F and d are one of the foci of the hyperbola and one of its directrixes, located on one side of the ordinate axis of the canonical coordinate system.

In fact, for example, for the focus F_2 and the directrix d_2 (Fig. 3.41, a) the condition \frac(r_2)(\rho_2)=e can be written in coordinate form:

\sqrt((x-c)^2+y^2)=e\left(x-\frac(a^2)(c)\right)

Getting rid of irrationality and replacing e=\frac(c)(a),~c^2-a^2=b^2, we arrive at the canonical hyperbola equation (3.50). Similar reasoning can be carried out for the focus F_1 and the directrix d_1:

\frac(r_1)(\rho_1)=e \quad \Leftrightarrow \quad \sqrt((x+c)^2+y^2)= e\left(x+\frac(a^2)(c) \right ).

Equation of a hyperbola in a polar coordinate system

The equation of the right branch of the hyperbola in the polar coordinate system F_2r\varphi (Fig. 3.41,b) has the form

R=\frac(p)(1-e\cdot\cos\varphi), where p=\frac(p^2)(a) - focal parameter of hyperbola.

In fact, let us choose the right focus F_2 of the hyperbola as the pole of the polar coordinate system, and the ray with the beginning at the point F_2, which belongs to the straight line F_1F_2, but does not contain the point F_1 (Fig. 3.41,b), as the polar axis. Then for an arbitrary point M(r,\varphi) belonging to the right branch of the hyperbola, according to the geometric definition (focal property) of the hyperbola, we have F_1M-r=2a. We express the distance between points M(r,\varphi) and F_1(2c,\pi) (see paragraph 2 of remarks 2.8):

F_1M=\sqrt((2c)^2+r^2-2\cdot(2c)^2\cdot r\cdot\cos(\varphi-\pi))=\sqrt(r^2+4\cdot c \cdot r\cdot\cos\varphi+4\cdot c^2).

Therefore, in coordinate form, the hyperbola equation has the form

\sqrt(r^2+4\cdot c\cdot r\cdot\cos\varphi+4\cdot c^2)-r=2a.

We isolate the radical, square both sides of the equation, divide by 4 and present similar terms:

R^2+4cr\cdot\cos\varphi+4c^2=4a^2+4ar+r^2 \quad \Leftrightarrow \quad a\left(1-\frac(c)(a)\cos\varphi\ right)r=c^2-a^2.

Express the polar radius r and make substitutions e=\frac(c)(a),~b^2=c^2-a^2,~p=\frac(b^2)(a):

R=\frac(c^2-a^2)(a(1-e\cos\varphi)) \quad \Leftrightarrow \quad r=\frac(b^2)(a(1-e\cos\varphi) )) \quad \Leftrightarrow \quad r=\frac(p)(1-e\cos\varphi),

Q.E.D. Note that in polar coordinates the equations of a hyperbola and an ellipse coincide, but describe different lines, since they differ in eccentricities ( e>1 for a hyperbola, 0\leqslant e<1 для эллипса).

Geometric meaning of the coefficients in the hyperbola equation

Let's find the points of intersection of the hyperbola (Fig. 3.42,a) with the abscissa axis (the vertices of the hyperbola). Substituting y=0 into the equation, we find the abscissa of the intersection points: x=\pm a. Therefore, the vertices have coordinates (-a,0),\,(a,0) . The length of the segment connecting the vertices is 2a. This segment is called the real axis of the hyperbola, and the number a is the real semi-axis of the hyperbola. Substituting x=0, we get y=\pm ib. The length of the y-axis segment connecting the points (0,-b),\,(0,b) is equal to 2b. This segment is called the imaginary axis of the hyperbola, and the number b is the imaginary semi-axis of the hyperbola. A hyperbola intersects the line containing the real axis, but does not intersect the line containing the imaginary axis.

Notes 3.10.

1. The straight lines x=\pm a,~y=\pm b limit the main rectangle on the coordinate plane, outside of which the hyperbola is located (Fig. 3.42, a).

2. Straight lines containing the diagonals of the main rectangle are called asymptotes of the hyperbola (Fig. 3.42, a).

For equilateral hyperbola described by the equation (i.e. for a=b), the main rectangle is a square whose diagonals are perpendicular. Therefore, the asymptotes of an equilateral hyperbola are also perpendicular, and they can be taken as the coordinate axes of the rectangular coordinate system Ox"y" (Fig. 3.42, b). In this coordinate system, the hyperbola equation has the form y"=\frac(a^2)(2x")(the hyperbola coincides with the graph of an elementary function expressing an inversely proportional relationship).

Indeed, let us rotate the canonical coordinate system by an angle \varphi=-\frac(\pi)(4)(Fig. 3.42, b). In this case, the coordinates of the point in the old and new coordinate systems are related by the equalities

\left\(\!\begin(aligned)x&=\frac(\sqrt(2))(2)\cdot x"+\frac(\sqrt(2))(2)\cdot y",\\ y& =-\frac(\sqrt(2))(2)\cdot x"+\frac(\sqrt(2))(2)\cdot y"\end(aligned)\right \quad \Leftrightarrow \quad \ left\(\!\begin(aligned)x&=\frac(\sqrt(2))(2)\cdot(x"+y"),\\ y&=\frac(\sqrt(2))(2) \cdot(y"-x")\end(aligned)\right.

Substituting these expressions into Eq. \frac(x^2)(a^2)-\frac(y^2)(a^2)=1 equilateral hyperbola and bringing similar terms, we get

\frac(\frac(1)(2)(x"+y")^2)(a^2)-\frac(\frac(1)(2)(y"-x")^2)(a ^2)=1 \quad \Leftrightarrow \quad 2\cdot x"\cdot y"=a^2 \quad \Leftrightarrow \quad y"=\frac(a^2)(2\cdot x").

3. The coordinate axes (of the canonical coordinate system) are the axes of symmetry of the hyperbola (called the main axes of the hyperbola), and its center is the center of symmetry.

Indeed, if the point M(x,y) belongs to the hyperbola . then the points M"(x,y) and M""(-x,y), symmetrical to the point M with respect to the coordinate axes, also belong to the same hyperbola.

The axis of symmetry on which the foci of the hyperbola are located is the focal axis.

4. From the hyperbola equation in polar coordinates r=\frac(p)(1-e\cos\varphi)(see Fig. 3.41, b) the geometric meaning of the focal parameter is clarified - this is half the length of the chord of the hyperbola passing through its focus perpendicular to the focal axis ( r = p at \varphi=\frac(\pi)(2)).

5. Eccentricity e characterizes the shape of the hyperbola. The larger e, the wider the branches of the hyperbola, and the closer e is to one, the narrower the branches of the hyperbola (Fig. 3.43, a).

Indeed, the value \gamma of the angle between the asymptotes of the hyperbola containing its branch is determined by the ratio of the sides of the main rectangle: \operatorname(tg)\frac(\gamma)(2)=\frac(b)(2). Considering that e=\frac(c)(a) and c^2=a^2+b^2 , we get

E^2=\frac(c^2)(a^2)=\frac(a^2+b^2)(a^2)=1+(\left(\frac(b)(a)\right )\^2=1+\operatorname{tg}^2\frac{\gamma}{2}. !}

The larger e, the larger the angle \gamma. For an equilateral hyperbola (a=b) we have e=\sqrt(2) and \gamma=\frac(\pi)(2). For e>\sqrt(2) the angle \gamma is obtuse, and for 1

6. Two hyperbolas defined in the same coordinate system by the equations \frac(x^2)(a^2)-\frac(y^2)(b^2)=1 and are called linked to each other. Conjugate hyperbolas have the same asymptotes (Fig. 3.43b). Equation of the conjugate hyperbola -\frac(x^2)(a^2)+\frac(y^2)(b^2)=1 reduced to canonical by renaming the coordinate axes (3.38).

7. Equation \frac((x-x_0)^2)(a^2)-\frac((y-y_0)^2)(b^2)=1 defines a hyperbola with center at point O"(x_0,y_0), the axes of which are parallel to the coordinate axes (Fig. 3.43, c). This equation is reduced to the canonical one using parallel translation (3.36). Equation -\frac((x-x_0)^2)(a^2)+\frac((y-y_0)^2)(b^2)=1 defines the conjugate hyperbola with center at point O"(x_0,y_0) .

Parametric hyperbola equation

The parametric equation of a hyperbola in the canonical coordinate system has the form

\begin(cases)x=a\cdot\operatorname(ch)t,\\y=b\cdot\operatorname(sh)t,\end(cases)t\in\mathbb(R),

Where \operatorname(ch)t=\frac(e^t+e^(-t))(2)- hyperbolic cosine, a \operatorname(sh)t=\frac(e^t-e^(-t))(2) hyperbolic sine.

Indeed, substituting the coordinate expressions into equation (3.50), we arrive at the main hyperbolic identity \operatorname(ch)^2t-\operatorname(sh)^2t=1.

Example 3.21. Draw a hyperbole \frac(x^2)(2^2)-\frac(y^2)(3^2)=1 in the canonical coordinate system Oxy. Find the semi-axes, focal length, eccentricity, focal parameter, equations of asymptotes and directrixes.

Solution. Comparing the given equation with the canonical one, we determine the semi-axes: a=2 - real semi-axis, b=3 - imaginary semi-axis of the hyperbola. We build the main rectangle with sides 2a=4,~2b=6 with the center at the origin (Fig. 3.44). We draw asymptotes by extending the diagonals of the main rectangle. We construct a hyperbola, taking into account its symmetry with respect to the coordinate axes. If necessary, determine the coordinates of some points of the hyperbola. For example, substituting x=4 into the hyperbola equation, we get

\frac(4^2)(2^2)-\frac(y^2)(3^2)=1 \quad \Leftrightarrow \quad y^2=27 \quad \Leftrightarrow \quad y=\pm3\sqrt (3).

Therefore, the points with coordinates (4;3\sqrt(3)) and (4;-3\sqrt(3)) belong to the hyperbola. Calculating the focal length

2\cdot c=2\cdot\sqrt(a^2+b^2)=2\cdot\sqrt(2^2+3^2)=2\sqrt(13)

eccentricity e=\frac(c)(a)=\frac(\sqrt(13))(2); focal parameter p=\frac(b^2)(a)=\frac(3^2)(2)=4,\!5. We compose the equations of asymptotes y=\pm\frac(b)(a)\,x, that is y=\pm\frac(3)(2)\,x, and the directrix equations: x=\pm\frac(a^2)(c)=\frac(4)(\sqrt(13)).

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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.

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 general case This number doesn't 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:

In practice, a combination of rotation by an arbitrary angle and parallel translation of the hyperbola is often encountered. This situation is discussed in class Reducing the 2nd order line equation to canonical form.

Parabola and its canonical equation

It's finished! She's the one. Ready to reveal many secrets. The canonical equation of a parabola has the form , where is a real number. It is easy to notice that in its standard position the parabola “lies on its side” and its vertex is at the origin. In this case, the function specifies the upper branch of this line, and the function – the lower branch. It is obvious that the parabola is symmetrical about the axis. Actually, why bother:

Example 6

Construct a parabola

Solution: the vertex is known, let’s find additional points. Equation determines the upper arc of the parabola, the equation determines the lower arc.

In order to shorten the recording of the calculations, we will carry out the calculations “with one brush”:

For compact recording, the results could be summarized in a table.

Before performing an elementary point-by-point drawing, let’s formulate a strict

definition of parabola:

A parabola is the set of all points in the plane that are equidistant from a given point and a given line that does not pass through the point.

The point is called focus parabolas, straight line - headmistress (spelled with one "es") parabolas. The constant "pe" of the canonical equation is called focal parameter, which is equal to the distance from the focus to the directrix. In this case. In this case, the focus has coordinates , and the directrix is ​​given by the equation .
In our example:

The definition of a parabola is even simpler to understand than the definitions of an ellipse and a hyperbola. For any point on a parabola, the length of the segment (the distance from the focus to the point) is equal to the length of the perpendicular (the distance from the point to the directrix):

Congratulations! Many of you have made a real discovery today. It turns out that a hyperbola and a parabola are not graphs of “ordinary” functions at all, but have a pronounced geometric origin.

Obviously, as the focal parameter increases, the branches of the graph will “raise” up and down, approaching infinitely close to the axis. As the “pe” value decreases, they will begin to compress and stretch along the axis

The eccentricity of any parabola is equal to unity:

Rotation and parallel translation of a parabola

The parabola is one of the most common lines in mathematics, and you will have to build it really often. Therefore, please pay special attention to the final paragraph of the lesson, where I will discuss typical options for the location of this curve.

! Note : as in the cases with previous curves, it is more correct to talk about rotation and parallel translation of coordinate axes, but the author will limit himself to a simplified version of the presentation so that the reader has a basic understanding of these transformations.

Presentation and lesson on the topic:
"Hyperbole, definition, property of a function"

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Hyperbole, definition

Guys, today we will study a new function and build its graph.
Consider the function: $y=\frac(k)(x)$, $k≠0$.
Coefficient $k$ – can take any real value except zero. For simplicity, let's start analyzing the function from the case when $k=1$.
Let's plot the function: $y=\frac(1)(x)$.
As always, let's start by building a table. True, this time we will have to divide our table into two parts. Let's consider the case when $x>0$.
We need to mark six points with coordinates $(x;y)$, which are given in the table and connect them with a line.
Now let's see what we get with negative x. Let's do the same thing, mark the points and connect them with a line. We have built two pieces of the graph, let's combine them.

Graph of the function $y=\frac(1)(x)$.
The graph of such a function is called a “Hyperbola”.

Properties of a hyperbola

Agree, the graph looks pretty nice, and it is symmetrical about the origin. If we draw any straight line passing through the origin of coordinates from the first to the third quarter, then it will intersect our graph at two points that will be equally distant from the origin of coordinates.
A hyperbola consists of two parts, symmetrical about the origin. These parts are called branches of the hyperbola.
The branches of a hyperbola in one direction (left and right) tend more and more towards the x-axis, but never cross it. In the other direction (up and down) they tend to the ordinate axis, but will also never cross it (since it is impossible to divide by zero). In such cases, the corresponding lines are called asymptotes. The graph of a hyperbola has two asymptotes: the x-axis and the y-axis.

A hyperbola has not only a center of symmetry, but also an axis of symmetry. Guys, draw the straight line $y=x$ and see how our graph is divided. You can notice that if the part that is located above the straight line $y=x$ is superimposed on the part that is located below, then they will coincide, this means symmetry with respect to the straight line.

We have plotted the function $y=\frac(1)(x)$, but what will happen in the general case is $y=\frac(k)(x)$, $k>0$.
The graphs will be practically no different. The result will be a hyperbola with the same branches, only the more $k$, the farther the branches will be removed from the origin of coordinates, and the less $k$, the closer to the origin of coordinates.

For example, the graph of the function $y=\frac(10)(x)$ looks like this. The graph became “wider” and moved away from the origin.
But what about negative $k$? The graph of the function $y=-f(x)$ is symmetrical to the graph of $y=f(x)$ relative to the x-axis; you need to turn it upside down.
Let's take advantage of this property and plot the function $y=-\frac(1)(x)$.

Let us summarize the knowledge gained.
The graph of the function $y=\frac(k)(x)$, $k≠0$ is a hyperbola located in the first and third (second and fourth) coordinate quarters, for $k>0$ ($k

Properties of the function $y=\frac(k)(x)$, $k>0$

1. Domain of definition: all numbers except $x=0$.
2. $y>0$ for $x>0$, and $y 3. The function decreases on the intervals $(-∞;0)$ and $(0;+∞)$.



7. Range of values: $(-∞;0)U(0;+∞)$.

Properties of the function $y=\frac(k)(x)$, $k
1. Domain of definition: all numbers except $x=0$.
2. $y>0$ for $x 0$.
3. The function increases on the intervals $(-∞;0)$ and $(0;+∞)$.
4. The function is not limited either above or below.
5. The greatest and lowest values No.
6. The function is continuous on the intervals $(-∞;0)U(0;+∞)$ and has a discontinuity at the point $x=0$.
7. Range of values: $(-∞;0)U(0;+∞)$.

Hyperbola is a second-order plane curve that consists of two separate curves that do not intersect.
Hyperbole formula y = k/x, provided that k not equal 0 . That is, the vertices of the hyperbola tend to zero, but never intersect with it.

Hyperbola- this is a set of points on the plane, the modulus of the difference in distances from two points, called foci, is a constant value.

Properties:

1. Optical property: light from a source located in one of the focuses of the hyperbola is reflected by the second branch of the hyperbola in such a way that the extensions of the reflected rays intersect at the second focus.
In other words, if F1 and F2 are the foci of the hyperbola, then the tangent at any point X of the hyperbola is the bisector of the angle ∠F1XF2.

2. For any point lying on a hyperbola, the ratio of the distances from this point to the focus to the distance from the same point to the directrix is ​​a constant value.

3. Hyperbole has mirror symmetry about the real and imaginary axes, and also rotational symmetry when rotated through an angle of 180° around the center of the hyperbola.

4. Each hyperbole has conjugate hyperbola, for which the real and imaginary axes change places, but the asymptotes remain the same.

Properties of a hyperbola:

1) A hyperbola has two axes of symmetry (the main axes of the hyperbola) and a center of symmetry (the center of the hyperbola). In this case, one of these axes intersects with the hyperbola at two points, called the vertices of the hyperbola. It is called the real axis of the hyperbola (axis Oh for the canonical choice of the coordinate system). The other axis does not have common points with a hyperbola and is called its imaginary axis (in canonical coordinates - the axis Oh). On both sides of it are the right and left branches of the hyperbola. The foci of a hyperbola are located on its real axis.

2) The branches of the hyperbola have two asymptotes, determined by the equations

3) Along with hyperbola (11.3), we can consider the so-called conjugate hyperbola, defined by the canonical equation

for which the real and imaginary axis are swapped while maintaining the same asymptotes.

4) Eccentricity of the hyperbola e> 1.

5) Distance ratio r i from hyperbola point to focus F i to the distance d i from this point to the directrix corresponding to the focus is equal to the eccentricity of the hyperbola.

42. Hyperbole is the set of points in the plane for which the modulus of the difference in distances to two fixed points is F 1 and F 2 of this plane, called tricks, is a constant value.

Let us derive the canonical equation of a hyperbola by analogy with the derivation of the equation of an ellipse, using the same notation.

|r 1 - r 2 | = 2a, from where If we denote b² = c² - a², from here you can get

- canonical hyperbola equation. (11.3)

The locus of points for which the ratio of the distance to the focus and to a given straight line, called the directrix, is constant and greater than one is called a hyperbola. The given constant is called the eccentricity of the hyperbola

Definition 11.6.Eccentricity a hyperbola is called a quantity e = c/a.

Eccentricity:

Definition 11.7.Headmistress D i hyperbola corresponding to the focus F i, is called a straight line located in the same half-plane with F i relative to the axis Oh perpendicular to the axis Oh at a distance a/e from the origin.

43. The case of a conjugate, degenerate hyperbola (NOT COMPLETELY)

Each hyperbole has conjugate hyperbola, for which the real and imaginary axes change places, but the asymptotes remain the same. This corresponds to the replacement a And b on top of each other in a formula describing a hyperbola. The conjugate hyperbola is not the result of rotating the initial hyperbola through an angle of 90°; both hyperbolas differ in shape.

If the asymptotes of a hyperbola are mutually perpendicular, then the hyperbola is called equilateral . Two hyperbolas that have common asymptotes, but with the transverse and conjugate axes rearranged, are called mutually conjugate .