Hyperbola and its properties
Lecture notes 14.
Hyperbola and parabola and their properties. Equations of an ellipse, hyperbola and parabola in a polar coordinate system.
Literature.§ 20, 21.
Definition 1. It is customary to call a hyperbola a set of points on a plane, for each of which the modulus of the difference between the distances to two fixed points and belonging to the same plane is a constant value less than the distance between points and.
Points and, as in the case of an ellipse, will be called tricks... Obviously, one should assume that the focuses do not coincide with each other. Let, and the modulus of the difference between the distances from the point of the hyperbola to the foci is. Then, as follows from the definition
From the inequalities connecting the sides of the triangle, it follows that there are no points M for which. Note that this difference is equal if and only if M lies on a straight line, and does not belong to the segment between foci. We will also assume that a ¹ 0, otherwise the points satisfying this condition form the midpoint perpendicular of the segment.
Let us derive the hyperbola equation. As in the case of an ellipse, we introduce a rectangular Cartesian coordinate system, which will also be called canonical, the abscissa axis of which contains the foci and, and the ordinate axis coincides with the middle perpendicular of the segment (Fig. 67). In this system, the coordinates of the focuses are:. A point if and only if lies on a hyperbola when its coordinates satisfy the equation:
Let's simplify this equation. Let us expand the module:, and “unify” one of the radicals:. Let's square both sides of the resulting equation:
After simplifications, we get:. Let's square both parts again:, or
In view of inequality (17.1), in this connection there exists a number b, for which
Then. Dividing both sides of this equality by, we finally get:
Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, the coordinates of any point of the hyperbola satisfy equation (17.4). Let us show the opposite. Take an arbitrary point, the coordinates of which are the solution to this equation. Let be. These numbers will be called focal radii point M. We should show that. Equation (17.4) implies that
Since, then, replacing y in this expression by formula (17.6), we get:
From formula (17.3) it follows that. For this reason. Thus,
It is shown similarly that
Let's expand the modules in the resulting formulas. Let be. Then, in this regard. It follows from inequality (17.5) that. Since, multiplying these inequalities, we get:. Hence it follows that. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, and.
Let be. Then and. From inequality (17.5) it follows that, multiplying it with the inequality, we obtain: or. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, and. In both the first and second cases, the modulus of the focal radius difference is constant and equal. Equation (17.4) is a hyperbola equation. It bears the name canonical.
Consider the properties of a hyperbola that will allow us to construct its image. First, we find its intersection points with the axes of the canonical coordinate system. Let the point serve as the intersection point of the hyperbola with the abscissa axis. Then it follows from equation (17.4) that, ᴛ.ᴇ. or either. The hyperbola intersects the abscissa at two points:. It does not cross the ordinate. Indeed, if the point lies on the hyperbola, then the number satisfies the equation:, ĸᴏᴛᴏᴩᴏᴇ has no real roots. Points and are called peaks hyperbole, and numbers a and b- her real and imaginary semiaxes .
If a point lies on a hyperbola, then, as follows from its canonical equation, points and also lie on a hyperbola. It follows from this that the hyperbola is symmetric with respect to the axes and centrally symmetric with respect to the origin of the canonical coordinate system. For this reason, it is sufficient to construct the points of the hyperbola lying in the first coordinate quarter, and then reflect them symmetrically about the axes and the origin of the coordinate system. From formula (17.6) it follows that in this quarter the hyperbola coincides with the graph of the function. By means of mathematical analysis it is proved that at this function is continuous, smooth and increasing. At the same time, it has an asymptote. As it is proved in the course of mathematical analysis, the straight line if and only if it serves as the asymptote of the function at, when In this case
Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, direct -asymptote of the hyperbola in the first coordinate quarter. Since the hyperbola is symmetric with respect to the coordinate axes, the same line serves as its asymptote in the third quarter, and the line serves as its asymptote in the second and fourth quarters. The hyperbola is shown in Figure 67.
Let us indicate the method of constructing the points of the hyperbola with a compass and a line-eye. Let and be its foci, and be the points of intersection with the abscissa axis. Construct a circle a centered at a point of radius r... Next, we increase the opening of the compass by the length of the segment and construct a circle b centered at a point with a radius. It is clear that the intersection points of the circles a and b lie on the hyperbola. Changing the radius r you can build any number of points of the hyperbola (Fig. 68).
A hyperbola, like an ellipse, has a directory property.
Definition 2. The eccentricity of a hyperbola is usually understood as a number equal to:
From inequality (17.1) it follows that for a hyperbola (compare, for an ellipse, the eccentricity is less than one). Let us find out how the form of the hyperbola changes, if its eccentricity takes values from 1 to + .. Then from the formula (17.9) we get:. Let e ® 1; then a ® c... As we have already noted, in this case the hyperbola is "compressed", its branches approaching two rays of the abscissa axis, the origins of which lie at its foci. At a ® 0, the branches of the hyperbola "straighten" to the midpoint perpendicular of the segment, ᴛ.ᴇ. to the ordinate axis.
Definition 3.Straight lines defined by equations:
are called directrix hyperbola.
The headmistress is considered to correspond to the focus, and to the focus. Since, then. For this reason, the directrixes intersect the abscissa axis at the interior points of the segment enclosed between the vertices of the hyperbola (Fig. 69). Let us prove the directory hyperbola property.
Theorem. A hyperbola is a set of all points of the plane, for each of which the ratio of the distance from this point to the focus to the distance to the directrix corresponding to this focus is a constant number equal to the eccentricity.
Proof. Let a hyperbola be given. We will assume that the plane has its canonical coordinate system. Consider a point lying on a hyperbola. Let us denote by and its distances to directrices and. From the formula for calculating the distance from a point to a straight line (see § 14) it follows that,. Let us find the ratios and, where and are the focal radii of the point M... From equalities (17.7) - (17.9), we obtain: and. For this reason.
Let us show the opposite. Let the ratio of the distance from some point M to the focus of the hyperbola to the distance from it to the corresponding directrix is equal to the eccentricity. Let us check that the point lies on the hyperbola. We carry out the proof for the focus and the directrix. For the second foci and directrix, the reasoning is similar. Let the coordinates of the point be given:. Then. The distance to the director is equal to:. Since, then. From here
Since (see (17.3)), then, or. Point M belongs to the hyperbola, the theorem is proved.
The directory properties of the ellipse and hyperbola allow a different approach to the definition of these curves. It follows from the theorems proved that if a straight line (directrix) and a point (focus) that does not lie on this straight line are given on the plane, then the set of all points of the plane, for each of which the ratio of the distance to the focus to the distance to the directrix, is constant number, represents an ellipse, if this number is less than one, and a hyperbola, if it is greater than one. The answer to the question of what form this set has, if the ratio is equal to one, will be given in the next section.
Let us answer the question, what form does the set of points have, for each of which the ratio of the distance to a point to the distance to a straight line that does not contain this point is equal to one. We will show that such a set of points is well known from the school algebra course; it coincides with a parabola.
Definition 1. The set of points on the plane, for each of which the distance to a fixed point of the plane is equal to the distance to a fixed line that does not contain this point, is called a parabola.
The point and the line, which are mentioned in the definition, will be called, respectively. focus and headmistress parabolas. We will also assume that the eccentricity of the parabola is equal to one. It is easy to find out what a set of points that satisfy Definition 1 is, if the focus lies on the directrix. If F- focus, d- the headmistress, and M is a point of the set, then in this case the segment FM perpendicular d... For this reason, such a set coincides with a straight line passing through the focus perpendicular to the directrix.
Let us derive the equation of the parabola. To do this, select a rectangular Cartesian coordinate system so that the abscissa axis passes through the focus F and was perpendicular to darektrix d parabola, and its beginning O coincided with the midpoint of the line between F and point Q the intersection of the abscissa and directrix axes. The direction of the abscissa axis is determined by a vector (Fig. 71). Such a coordinate system will be called canonical... Let us denote by p segment length FQ, Number R it is customary to call focal parameter parabolas. Then, in the canonical system, the focus coordinates F and the directrix equation d has the form:,
Consider an arbitrary point. distance R from M before F equals: . The length of the perpendicular d omitted from M to the headmistress d, according to the formula for calculating the distance from a point to a straight line (see § 14), has the form:. For this reason, from Definition 1 it follows that point M if and only if lies on a parabola when
Equation (18.1) is the equation of a parabola. It is imperative for us to simplify it. To do this, let's square both sides:
Hence it follows that
After bringing similar terms, we get:
Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, if a point belongs to a parabola, then its coordinates satisfy equation (18.4). It is easy to see the opposite. If the coordinates of the point M serve as a solution to equation (18.4), then they satisfy equations (18.3) and (18.2). Taking the square root from both sides of equality (18.2), we obtain that the coordinates of the point M satisfy (18.1). The point lies on the parabola.
Equation (18.4) is called canonical equation parabolas. Let us note its properties. Start O of the canonical coordinate system lies on the parabola, since it is a solution to equation (18.4). It is customary to call it the pinnacle. The parabola is symmetrical about the abscissa axis and not symmetrical about the ordinate axis of the canonical system. Indeed, if the coordinates of a point satisfy equation (18.4), then the coordinates of the point also satisfy equation (18.4), and the coordinates of the point are not a solution to this equation. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, to build a parabola, it is enough to display the graph of the power function, and then display it symmetrically about the abscissa axis. By means of mathematical analysis it is proved that it is a continuous, smooth and infinitely increasing function. The parabola is shown in Figure 71.
Consider a method for constructing the points of a parabola. Let be F- her focus, and d- the headmistress. Let us draw the axis of symmetry of the parabola, ᴛ.ᴇ. straight l containing F and perpendicular d... Next, we will construct several straight lines perpendicular to the axis. On each straight line, we define two points of intersection with a circle, the center of which is in focus F, and the radius is equal to the distance between this straight line and the directrix (see Fig. 72). It is clear that these points lie on a parabola.
Let the curve g be an ellipse, one branch of a hyperbola, or a parabola. Let be F- focus, and d is the directrix of the g curve corresponding to this focus. In this case, we will assume that, in the case of a hyperbola, the focus and directrix are chosen so that the considered branch of the curve lies in the same half-plane with respect to d as the focus F... We will also assume that the pole of the polar coordinate system coincides with F, and the polar axis l- lies on the axis of symmetry and does not intersect the directrix d (Fig. 74). Raise at point F the perpendicular to l, R is the point of its intersection with γ. Let us denote by R segment length FР... Number R will be called the focal parameter g.
Let r and j denote the polar coordinates of the point M... Recall that in our case, and j is the oriented angle between the polar axis l and vector. Let us denote by Q and N point projections R and M to the headmistress d and after TO- projection M on the axis of symmetry of the curve g (see Fig. 74). Then, if R- point of intersection of the directrix d and the axis of symmetry l, then Since the projection on l has the form:, a, then. Let us use the directory property of the second-order curve. If e is the eccentricity of g, then. For this reason, as well. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ,. Multiplying this ratio by e and highlighting r, we finally get:
Equation (18.6) is usually called polar equation curve of the second order g.
Let e< 1. Тогда g представляет собой эллипс. В этом случае для любого j: . Так как полярный радиус всегда положителен, то для любого угла φ существует значение, ρ определяемое формулой (18.6), для которого точка M(r; j) lies on an ellipse. Any ray with the origin at the pole of the polar coordinate system intersects the ellipse (Fig. 75). If e = 1, then g is a - parabola. In this case, for any j:, and for j = 0. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, in equation (18.6) j takes all values on the half-interval (- p; p], with the exception of 0. Any ray with origin at the focus F, after excluding the polar axis, intersects the parabola (Fig. 76). Consider the case when e> 1. Then g is a branch of the hyperbola. As follows from equation (18.6), the angle j satisfies the inequality.
Let's solve this inequality. Let be. Since, then. We use the formulas expressing the eccentricity of the hyperbola through its semiaxes and the distance between the foci (see § 17), we get:, ᴛ.ᴇ. ... It is easy to see that j is a solution to inequality (18.7) if and only if,. Geometrically, this means that if the angle φ belongs to the segment [; ], then the ray making an angle j with the polar axis and with the origin at focus F does not intersect the branch of the hyperbola. Note that the rays forming angles with the polar axis equal to and are parallel to the asymptotes of the hyperbola (Fig. 77). It can be proved that if generalized polar coordinates are introduced on the plane (see § 9), then equation (18.6) in the case defines the second branch of the hyperbola.
Hyperbola and its properties - concept and types. Classification and features of the category "Hyperbola and its properties" 2017, 2018.
Hyperbole is called the locus of points of the plane, the coordinates of which satisfy the equation
Hyperbola parameters:
Points F 1 (–c, 0), F 2 (c, 0), where are called tricks hyperbole, while the value 2 with (with > a> 0) determines focal length ... Points A 1 (–a, 0), A 2 (a, 0) are called the vertices of the hyperbola , wherein A 1 A 2 = 2a forms real axis hyperbole, and V 1 V 2 = 2b – imaginary axis ( V 1 (0, –b), B 2 (0, b)), O – Centre hyperbole.
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The magnitude called eccentricity hyperbole, it characterizes the degree of "compaction" of hyperbole;
– focal radii
hyperbole (point M belongs to the hyperbola), and r 1 = a + εx, r 2 = –a + εx for points of the right branch of the hyperbola, r 1 = – (a + εx), r 2 = – (–a + εx) - for points of the left branch;
– headmistresses
hyperbole;
– asymptote equations .
For hyperbole it is true: ε
> 1, the directresses do not cross the border and the inner region of the hyperbola, and also have the property 
They say that the equation
asks the equation of the conjugate hyperbola (fig. 20). It can also be written as
In this case, the axis is imaginary, the foci lie on the axis. All other parameters are determined in the same way as for hyperbola (25).
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The points of the hyperbola have an important characteristic property: the absolute value of the difference in distance from each of them to the foci is a constant value equal to 2 a(fig. 19).
For parametric assignment hyperbolas as a parameter t the value of the angle between the radius vector of the point lying on the hyperbola and the positive direction of the axis can be taken Ox:
Example 1. Give the equation of hyperbola
9x 2 – 16y 2 = 144
to the canonical form, find its parameters, depict a hyperbola.
Solution. We divide the left and right sides of the given equation by 144: From the last equation it follows directly: a = 4, b = 3, c = 5, O(0, 0) is the center of the hyperbola. Focuses are in points F 1 (–5, 0) and F 2 (5, 0), eccentricity ε = 5/4, directors D 1 and D 2 are described by the equations D 1: x = –16/5, D 2: x= 16/5, asymptotes l 1 and l 2 have equations
Let's make a drawing. To do this, along the axes Ox and Oy symmetrically with respect to the point (0, 0) set aside the segments A 1 A 2 = 2a= 8 and V 1 V 2 = 2b= 6 respectively. Through the obtained points A 1 (–4, 0), A 2 (4, 0), V 1 (0, –3), V 2 (0, 3) draw straight lines parallel to the coordinate axes. As a result, we get a rectangle (Fig. 21), the diagonals of which lie on the asymptotes of the hyperbola. Building a hyperbola
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To find the angle φ between the asymptotes of the hyperbola, we use the formula
.
,
whence we get 
Example 2 . Determine the type, parameters and location on the plane of the curve, the equation of which
Solution. Let us simplify the right side of this equation using the method of extracting perfect squares:
We get the equation
which by dividing by 30 is reduced to the form
This is the equation of a hyperbola, the center of which lies at the point the real semiaxis - the imaginary semiaxis - (Fig. 22).
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Example 3. Make the equation of the hyperbola, conjugate with respect to the hyperbola, determine its parameters and make a drawing.
Solution. The equation of the conjugate hyperbola is
Real semiaxis b= 3, imaginary - a= 4, half of the focal distance The vertices of the hyperbola are the points B 1 (0, –3) and V 2 (0, 3); her focuses are in points F 1 (0, –5) and F 2 (0, 5); eccentricity ε = with/b= 5/3; headmistresses D 1 and D 2 are given by the equations D 1: y = –9/5, D 2: y= 9/5; the equations are the equations of the asymptotes (Fig. 23).
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Note that the common elements for conjugate hyperbolas are the auxiliary "rectangle" and the asymptotes.
Example 4. Write the equation of the hyperbola with semiaxes a and b (a > 0, b> 0) if it is known that its principal axes are parallel to the coordinate axes. Determine the main parameters of the hyperbole.
Solution. The desired equation can be considered as a hyperbola equation, which is obtained as a result of the parallel transfer of the old coordinate system to the vector where ( x 0 , y 0) is the center of the hyperbola in the "old" coordinate system. Then, using the relations between the coordinates of an arbitrary point M plane in the given and transformed systems
Hyperbola is a plane curve, for each point of which the modulus of the difference between the distances to two given points ( foci of hyperbole
) is constant. The distance between the foci of the hyperbola is called focal length
and is denoted by \ (2c \). The middle of the segment connecting the foci is called center... The hyperbola has two axes of symmetry: the focal or real axis, passing through the foci, and the imaginary axis perpendicular to it, passing through the center. The real axis crosses the branches of the hyperbola at points called peaks... The segment connecting the center of the hyperbola with the vertex is called real semiaxis
and is denoted by \ (a \). Imaginary semiaxis
denoted by \ (b \). Canonical hyperbola equation
written in the form
\ (\ large \ frac (((x ^ 2))) (((a ^ 2))) \ normalsize - \ large \ frac (((y ^ 2))) (((b ^ 2))) \ normalsize = 1 \).
The modulus of the difference in distances from any point of the hyperbola to its foci is constant:
\ (\ left | ((r_1) - (r_2)) \ right | = 2a \),
where \ ((r_1) \), \ ((r_2) \) - distances from an arbitrary point \ (P \ left ((x, y) \ right) \) hyperbola to foci \ ((F_1) \) and \ ( (F_2) \), \ (a \) is the real semiaxis of the hyperbola.
Equations of the asymptotes of the hyperbola
\ (y = \ pm \ large \ frac (b) (a) \ normalsize x \)
Relationship between semi-axes of hyperbola and focal length
\ ((c ^ 2) = (a ^ 2) + (b ^ 2) \),
where \ (c \) is half the focal length, \ (a \) is the real semiaxis of the hyperbola, \ (b \) is the imaginary semiaxis.
Eccentricity
hyperbole
\ (e = \ large \ frac (c) (a) \ normalsize> 1 \)
Directrix hyperbola equations
The directrix of a hyperbola is a straight line perpendicular to its real axis and intersecting it at a distance \ (\ large \ frac (a) (e) \ normalsize \) from the center. The hyperbole has two directresses, located on opposite sides of the center. The directrix equations are
\ (x = \ pm \ large \ frac (a) (e) \ normalsize = \ pm \ large \ frac (((a ^ 2))) (c) \ normalsize \).
Equation of the right branch of the hyperbola in parametric form
\ (\ left \ (\ begin (aligned) x & = a \ cosh t \\ y & = b \ sinh t \ end (aligned) \ right., \; \; 0 \ le t \ le 2 \ pi \ ),
where \ (a \), \ (b \) are the semiaxes of the hyperbola, \ (t \) is a parameter.
General equation of hyperbola
where \ (B ^ 2 - 4AC> 0 \).
General equation of a hyperbola whose semiaxes are parallel to the coordinate axes
\ (A (x ^ 2) + C (y ^ 2) + Dx + Ey + F = 0 \),
where \ (AC
Equilateral hyperbola
The hyperbola is called isosceles
if its semiaxes are the same: \ (a = b \). In such a hyperbola, the asymptotes are mutually perpendicular. If the asymptotes are the horizontal and vertical coordinate axes (respectively, \ (y = 0 \) and \ (x = 0 \)), then the equation of an isosceles hyperbola has the form
\ (xy = \ large \ frac (((e ^ 2))) (4) \ normalsize \) or \ (y = \ large \ frac (k) (x) \ normalsize \), where \ (k = \ large \ frac (e ^ 2) (4) \ normalsize. \)
Parabola is called a flat curve, at each point of which the following property is satisfied: the distance to a given point ( focus parabola
) is equal to the distance to a given straight line ( parabola directrix
). The distance from the focus to the directrix is called parabola parameter
and is denoted by \ (p \). The parabola has a single axis of symmetry that intersects the parabola at its the top
. Canonical parabola equation
has the form
\ (y = 2px \).
Directrix equation
\ (x = - \ large \ frac (p) (2) \ normalsize \),
Focus coordinates
\ (F \ left ((\ large \ frac (p) (2) \ normalsize, 0) \ right) \)
Vertex coordinates
\ (M \ left ((0,0) \ right) \)
General parabola equation
\ (A (x ^ 2) + Bxy + C (y ^ 2) + Dx + Ey + F = 0 \),
where \ (B ^ 2 - 4AC = 0 \).
Equation of a parabola whose symmetry axis is parallel to the \ (Oy \) axis
\ (A (x ^ 2) + Dx + Ey + F = 0 \; \ left ((A \ ne 0, E \ ne 0) \ right) \),
or in equivalent form
\ (y = a (x ^ 2) + bx + c, \; \; p = \ large \ frac (1) (2a) \ normalsize \)
Directrix equation
\ (y = (y_0) - \ large \ frac (p) (2) \ normalsize \),
where \ (p \) is the parameter of the parabola.
Focus coordinates
\ (F \ left (((x_0), (y_0) + \ large \ frac (p) (2) \ normalsize) \ right) \)
Vertex coordinates
\ ((x_0) = - \ large \ frac (b) ((2a)) \ normalsize, \; \; (y_0) = ax_0 ^ 2 + b (x_0) + c = \ large \ frac ((4ac - ( b ^ 2))) ((4a)) \ normalsize \)
Equation of a parabola with apex at the origin and an axis of symmetry parallel to the \ (Oy \) axis
\ (y = a (x ^ 2), \; \; p = \ large \ frac (1) ((2a)) \ normalsize \)
Directrix equation
\ (y = - \ large \ frac (p) (2) \ normalsize \),
where \ (p \) is the parameter of the parabola.
Focus coordinates
\ (F \ left ((0, \ large \ frac (p) (2) \ normalsize) \ right) \)
Vertex coordinates
\ (M \ left ((0,0) \ right) \)
For the rest of the readers, I propose to significantly expand their school knowledge about the parabola and hyperbola. Are hyperbola and parabola easy? ... 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 hyperbola and the problem of constructing it.
The canonical hyperbola equation 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 "bh".
I must say, rather unexpectedly ... the equation of the "school" hyperbole does not even come close to resembling the canonical notation. But this riddle will wait for us, but for now, scratch the back of your head and remember what characteristic features the curve in question has? Let's spread our imagination on the screen function graph ….
The hyperbola has two symmetrical branches.
Nice progress! Any hyperbole possesses 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: at the first step, we bring this equation to the canonical form. Please remember the typical procedure. On the right, you need to get "one", so we divide both sides of the original equation by 20: 
Here you can cancel both fractions, but it is more optimal to make each of them three-story:
And only after that carry out the reduction:
Select the squares in the denominators:
Why is the transformation better done this way? After all, the fractions of the left side can be immediately reduced and obtained. The fact is that in the example under consideration I was 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, an equation. Here, with divisibility, everything is sadder and without three-story fractions not enough: 
So, let's use the fruit of our labors - the canonical equation:
How to build 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 it is utopian, so it is much more profitable to re-involve 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 through an arbitrary angle and parallel translation of a hyperbola is often encountered. This situation is considered in the lesson. Reducing the equation of the second order line to the canonical form.
Parabola and its canonical equation
It is finished! She is the most. Ready to reveal many secrets. The canonical equation of the parabola has the form, where is a real number. It is easy to see that in its standard position the parabola "lies on its side" and its vertex is at the origin. In this case, the function sets the upper branch of the given line, and the function sets the lower branch. Obviously, the parabola is symmetrical about the axis. Actually, why bother:
Example 6
Construct parabola
Solution: the vertex is known, find additional points. The equation 
defines the upper arc of the parabola, the equation defines the lower arc.
In order to shorten the recording of the calculation, we will carry out "under one comb": 
For a compact record, the results could be tabulated.
Before performing an elementary pointwise drawing, we formulate a strict
definition of a parabola:
A parabola is the set of all points of the plane that are equidistant from a given point and a given straight line that does not pass through the point.
The point is called focus parabolas, straight - headmistress (written with one "es") parabolas. The constant "ne" 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 easier to understand than the definitions of an ellipse and a hyperbola. For any point of the 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 the hyperbola and parabola are not at all graphs of "ordinary" functions, but have a pronounced geometric origin.
Obviously, with an increase in the focal parameter, the branches of the graph will be "distributed" up and down, infinitely close to the axis. With a decrease in the value of "pe", they will begin to shrink and stretch along the axis
The eccentricity of any parabola is equal to one:
Rotation and parallel translation of a parabola
The parabola is one of the most common lines in mathematics, and you will have to draw it very often. Therefore, please, pay particular attention to the final paragraph of the lesson, where I will analyze the typical options for the location of this curve.
! Note : as in the cases with the previous curves, it is more correct to talk about rotation and parallel translation of the coordinate axes, but the author will limit himself to a simplified version of the presentation so that the reader has elementary ideas about these transformations.
Definition 7.2. The locus of points of the plane, for which the difference in distances to two fixed points is a constant value, is called hyperbole.
Remark 7.2. When talking about the difference in distances, it is assumed that the smaller one is subtracted from the greater distance. This means that, in fact, for a hyperbola, the modulus of the difference in distances from any of its points to two fixed points is constant. #
The definition of hyperbola is similar to the definition ellipse... The only difference between them is that for a hyperbola, the difference in distances to fixed points is constant, and for an ellipse, the sum of the same distances. Therefore, it is natural that these curves have much in common both in properties and in the terminology used.
Fixed points in the definition of hyperbola (denote them by F 1 and F 2) are called focuses of hyperbole... The distance between them (let us designate it as 2c) is called focal distance, and the segments F 1 M and F 2 M connecting an arbitrary point M on the hyperbola with its foci are focal radii.
The type of hyperbola is completely determined by the focal distance | F 1 F 2 | = 2c and the value of the constant 2a, equal to the difference between the focal radii, and its position on the plane is the position of the foci F 1 and F 2.
From the definition of a hyperbola, it follows that, like an ellipse, it is symmetric with respect to the straight line passing through the foci, as well as with respect to the straight line that divides the segment F 1 F 2 in half and is perpendicular to it (Fig. 7.7). The first of these axes of symmetry is called real axis of hyperbola, and the second is her imaginary axis... The constant value a participating in the definition of the hyperbola is called the real semi-axis of the hyperbola.
The midpoint of the segment F 1 F 2 connecting the foci of the hyperbola lies at the intersection of its axes of symmetry and therefore is the center of symmetry of the hyperbola, which is simply called center of hyperbole.
For a hyperbola, the real axis 2a should be no more than the focal distance 2c, since for the triangle F 1 MF 2 (see Fig. 7.7) the inequality || F 1 M | - | F 2 M | | ≤ | F 1 F 2 |. The equality a = c holds only for those points M that lie on the real axis of symmetry of the hyperbola outside the interval F 1 F 2. Discarding this degenerate case, we will further assume that a
Hyperbola equation... Consider on the plane some hyperbola with focuses at points F 1 and F 2 and a real axis 2a. Let 2c be the focal distance, 2c = | F 1 F 2 | > 2a. According to Remark 7.2, the hyperbola consists of those points M (x; y) for which | | F 1 M | - - | F 2 M | | = 2a. Let's choose rectangular coordinate system Oxy so that the center of the hyperbola is at origin, and the focuses were located on abscissa axis(fig. 7.8). Such a coordinate system for the considered hyperbola is called canonical, and the corresponding variables are canonical.
In the canonical coordinate system, the foci of the hyperbola have coordinates F 1 (c; 0) and F 2 (-c; 0). Using the formula for the distance between two points, we write down the condition || F 1 M | - | F 2 M || = 2а in coordinates | √ ((x - c) 2 + y 2) - √ ((x + c) 2 + y 2) | = 2a, where (x; y) are the coordinates of the point M. To simplify this equation, we get rid of the modulus sign: √ ((x - c) 2 + y 2) - √ ((x + c) 2 + y 2) = ± 2a, move the second radical to the right side and square it: (x - c) 2 + y 2 = (x + c) 2 + y 2 ± 4a √ ((x + c) 2 + y 2) + 4a 2 ... After simplification, we get -εx - a = ± √ ((x + c) 2 + y 2), or
√ ((x + c) 2 + y 2) = | εx + a | (7.7)
where ε = s / a. Let us square it again and again bring similar terms: (ε 2 - 1) x 2 - y 2 = c 2 - a 2, or, taking into account the equality ε = c / a and setting b 2 = c 2 - a 2,
x 2 / a 2 - y 2 / b 2 = 1 (7.8)
The quantity b> 0 is called imaginary semiaxis of hyperbola.
So, we have established that any point on the hyperbola with the foci F 1 (c; 0) and F 2 (-c; 0) and the real semiaxis a satisfies equation (7.8). But it is also necessary to show that the coordinates of points outside the hyperbola do not satisfy this equation. To do this, we consider the family of all hyperbolas with given foci F 1 and F 2. This family of hyperbolas has common axes of symmetry. From geometric considerations, it is clear that each point of the plane (except for points lying on the real axis of symmetry outside the interval F1F2, and points lying on the imaginary axis of symmetry) belongs to some hyperbola of the family, and only one, since the difference between the distances from the point to the foci F 1 and F 2 changes from hyperbole to hyperbole. Let the coordinates of the point M (x; y) satisfy Eq. (7.8), and the point itself belongs to the hyperbola of the family with some value г of the real semiaxis. Then, as we have proved, its coordinates satisfy the equation 
Consequently, the system of two equations with two unknowns
has at least one solution. By direct verification, we see that this is impossible for ã ≠ a. Indeed, excluding, for example, x from the first equation:
after transformations we obtain the equation
which has no solutions for ã ≠ a, since. So, (7.8) is a hyperbola equation with a real semiaxis a> 0 and an imaginary semiaxis b = √ (c 2 - a 2)> 0. It is called the canonical hyperbole equation.
Hyperbole view. In its form, the hyperbola (7.8) differs markedly from the ellipse. Taking into account the presence of two axes of symmetry of the hyperbola, it is enough to construct that part of it, which is in the first quarter of the canonical coordinate system. In the first quarter, i.e. for x ≥ 0, y ≥ 0, the canonical hyperbola equation is uniquely resolved with respect to y:
y = b / a √ (x 2 - a 2). (7.9)
Investigation of this function y (x) gives the following results.
The domain of the function is (x: x ≥ a) and in this domain it is continuous as a complex function, and at the point x = a it is right-continuous. The only zero of the function is the point x = a.
Let us find the derivative of the function y (x): y "(x) = bx / a√ (x 2 - a 2). Hence we conclude that for x> a the function increases monotonically. In addition, 
, and this means that at the point x = a of the intersection of the graph of the function with the abscissa, there is a vertical tangent. The function y (x) has a second derivative y "= -ab (x 2 - a 2) -3/2 for x> a, and this derivative is negative. Therefore, the graph of the function is convex upward, and there are no inflection points.
The indicated function has an oblique asymptote, this follows from the existence of two limits:
The oblique asymptote is described by the equation y = (b / a) x.
The conducted study of the function (7.9) allows us to construct its graph (Fig. 7.9), which coincides with the part of the hyperbola (7.8) contained in the first quarter.
Since the hyperbola is symmetric about its axes, the entire curve has the form shown in Fig. 7.10. The hyperbola consists of two symmetrical branches located at different
sides of its imaginary axis of symmetry. These branches are not bounded on both sides, and the straight lines y = ± (b / a) x are simultaneously the asymptotes of both the right and left branches of the hyperbola.
The symmetry axes of the hyperbola differ in that the real intersects the hyperbola, and the imaginary, being the locus of points equidistant from the foci, does not intersect (therefore, it is called imaginary). Two points of intersection of the real axis of symmetry with the hyperbola are called the vertices of the hyperbola (points A (a; 0) and B (-a; 0) in Fig. 7.10).
The construction of a hyperbola along its real (2a) and imaginary (2b) axes should begin with a rectangle centered at the origin and sides 2a and 2b, parallel, respectively, to the real and imaginary axes of symmetry of the hyperbola (Fig. 7.11). The asymptotes of the hyperbola are the continuation of the diagonals of this rectangle, and the vertices of the hyperbola are the intersection points of the sides of the rectangle with the real axis of symmetry. Note that the rectangle and its position on the plane uniquely determine the shape and position of the hyperbola. The b / a ratio of the sides of the rectangle determines the degree of compression of the hyperbola, but instead of this parameter, the eccentricity of the hyperbola is usually used. Eccentricity of hyperbola called the ratio of its focal distance to the real axis. Eccentricity is denoted by ε. For the hyperbola described by equation (7.8), ε = c / a. Note that if eccentricity of the ellipse can take values from a half-interval)