circumference is the collection of all points of the plane equidistant from one given point, called the center of the circle. The distance from the center of the circle to any point on the circle is called . circle radius.

- canonical equation circle(16) - the center of the circle.

If the center of the circle lies at the origin, then the circle equation (16 .)

Ellipse is called the set of all points of the plane, the sum of the distances of which from two given points of this plane (called tricks this ellipse) is a constant value.

In (0;b)M(x,y)

r 1 r 2 r 1 +r 2 =2a

(-а; 0) F 1 (-c; 0) 0 F 2 (c; 0) (а; 0) X

Let us denote for brevity a 2 -b 2 \u003d c 2 (*), then the ellipse equation: (17)

If we put y=0, then we get , and if we put x=0, we get ; hence, and are the lengths of the semi-axes of the ellipse - big() and small(). In addition, each of the terms on the left side cannot be greater than one, hence , , and therefore the entire ellipse is located inside the rectangle. Points A,B,C,D, in which the ellipse intersects its axes of symmetry, are called the vertices of the ellipse.

Attitude is called the eccentricity of the ellipse.

Hyperbole is the collection of all points of the plane, the modulus of the difference of the distances of which from two given points of this plane (called tricks this hyperbola) is a constant quantity. The middle of the distance between foci is called the center of the hyperbola.

r 2 r 1 –r 2 =2a

F 1 (-c; 0) 0 F 2 (c; 0) x

Denote a 2 -c 2 \u003d-b 2 (**), the hyperbola equation: (18)

It can be seen from this equation that the hyperbola also has two axes of symmetry (principal axes), as well as a center of symmetry (the center of the hyperbola).

Attitude is called the eccentricity of the hyperbola.

If we put y=0, then we get , and if we put x=0, we get .

So the Ox axis intersects the hyperbola at two points (the vertices of the hyperbola), this is - real axis; The Oy axis does not intersect the hyperbola - this is " imaginary axis. » Any segment connecting two points of a hyperbola, if it passes through the center, is called the diameter of the hyperbola.

A straight line to which a curved line approaches arbitrarily close, but never intersects it, is called curve asymptote. A hyperbola has two asymptotes. Their equations are: (19)

parabola called the set of all points of the plane, the distance from each of which to a given point (called focus) is equal to the distance to the given line (called headmistress).

- parabola parameter.

The parabola has one axis of symmetry. The point of intersection of the parabola with the axis of symmetry is called top of the parabola.

The canonical equation of a parabola with a vertex at the origin, whose axis of symmetry is the Ox axis and whose branches are directed to the right, has the form (20)

Its directrix equation is:

The canonical equation of a parabola with a vertex at the origin, whose axis of symmetry is the Ox axis and whose branches are directed to the left, has the form (20 ,)

Its directrix equation is:

The canonical equation of a parabola with a vertex at the origin, whose symmetry axis is the Oy axis and whose branches are directed upwards, has the form (20 ,)

Its directrix equation is:

The canonical equation of a parabola with a vertex at the origin, whose axis of symmetry is the Oy axis and whose branches are directed downwards, has the form (20 ,)

Its directrix equation is:

y y

F 0 p/2 x -p/2 0 x

Y y

p/2

-p/2
Topic 2.1. Lecture 7. Lesson 10

Topic: Functions of one independent variable, their graphs.

Function concept

One of the basic mathematical concepts is the concept of a function. The concept of a function is associated with the establishment of dependence (connection) between the elements of two sets.

Let two non-empty sets X and Y be given. The correspondence ƒ, which associates one and only one element yÎ Y with each element xÎ X, is called a function and is written y=ƒ(x), xÎ X or ƒ : X→Y. It is also said that the function ƒ maps the set X onto the set Y.

For example, the correspondences ƒ and g shown in Figure 98 a and b are functions, while those in Figure 98 c and d are not. In the case in - not every element xÎX corresponds to the element yÎY. In the case r, the uniqueness condition is not satisfied.

The set X is called the domain of the function ƒ and is denoted by D(f). The set of all унY is called the value set of the function ƒ and is denoted by E(ƒ).

Numeric functions. Function graph. Ways to set functions

Let a function ƒ : X→Y be given.

If the elements of the sets X and Y are real numbers (that is, XÌ R and YÌ R), then the function ƒ is called a number function. In the future, we will study (as a rule) numerical functions, for brevity we will call them simply functions and write y=ƒ(x).

The variable x is called an argument or independent variable, and y is called a function or dependent variable (of x). Regarding the values ​​x and y themselves, they say that they are in a functional relationship. Sometimes the functional dependence of y on x is written as y=y(x), without introducing a new letter (ƒ) to indicate the dependence.

private value functions ƒ(x) at x=a are written as follows: ƒ(a). For example, if ƒ(x)=2x 2 -3, then ƒ(0)=-3, ƒ(2)=5.

Function Graph y \u003d (x) is the set of all points of the Oxy plane, for each of which x is the value of the argument, and y is the corresponding value of the function.

For example, the graph of the function y \u003d √ (1-x 2) is the upper semicircle of radius R \u003d 1 with the center at O ​​(0; 0) (see Fig. 99).

To set the function y=ƒ(x), it is necessary to specify a rule that allows, knowing x, to find the corresponding value of y.

There are three most common ways to define a function: analytical, tabular, graphic.

Analytical method: The function is specified as one or more formulas or equations.

If the domain of the function y = ƒ(x) is not specified, then it is assumed that it coincides with the set of all values ​​of the argument for which the corresponding formula makes sense. So, the domain of the function y \u003d √ (1-x2) is the segment [-1; one].

The analytical method of setting the function is the most perfect, since it is accompanied by methods of mathematical analysis that allow you to fully explore the function y=ƒ(x).

Graphical method: the graph of the function is set.

Often graphs are drawn automatically by recorders or displayed on a display screen. The values ​​of the function y, corresponding to certain values ​​of the argument x, are directly found from this graph.

The advantage of a graphic task is its visibility, the disadvantage is its inaccuracy.

Tabular way: a function is specified by a table of a series of argument values ​​and corresponding function values. For example, the well-known tables of values trigonometric functions, logarithmic tables.

In practice, one often has to use tables of function values ​​obtained empirically or as a result of observations.

Consider the lines defined by the equation of the second degree with respect to the current coordinates

The coefficients of the equation are real numbers, but at least one of numbers A,B or C is different from 0. Such lines are called lines (curves) of the second order. Below we will show that equation (1) defines an Ellipse circle, a hyperbola, or a parabola in the plane.

Circle

The simplest curve of the second order is a circle. Recall that a circle of radius R centered at a point M 0 is a set of points M of the plane satisfying the condition MM 0 =R. Let the point M 0 in the Oxy system have coordinates x 0 ,y 0 , and M(x, y) be an arbitrary point of the circle. Then either

-canonical equation of a circle . Assuming x 0 \u003d y 0 \u003d 0 we get x 2 + y 2 \u003d R 2

let us show that the equation of a circle can be written as a general equation of the second degree (1). To do this, we square the right side of the circle equation and get:

In order for this equation to correspond to (1), it is necessary that:

1) coefficient B=0,

2). Then we get: (2)

The last equation is called the general equation of a circle . Dividing both sides of the equation by A ≠ 0 and adding the terms containing x and y to the full square, we get:

(2)

Comparing this equation with the canonical equation of a circle, we find that equation (2) is really an equation of a circle if:

1)A=C, 2)B=0, 3)D 2 +E 2 -4AF>0.

When these conditions are met, the center of the circle is located at the point O, and its radius .

Ellipse

y

x

F 2 (c, o)

F1 (-c,o)

By definition, 2 > 2c, that is, > c. To derive the ellipse equation, we assume that the foci F 1 and F 2 lie on the Ox axis, and t. O coincides with the midpoint of the segment F 1 F 2, then F 1 (-c, 0), F 2 (c,0).

Let M(x,y) be an arbitrary point of the ellipse, then, according to the definition of the ellipse, MF 1 +MF 2 =2, i.e.

This is the equation of the ellipse. You can change it to a simpler form like this:

Let's square it:

square up

Since , then 2 -c 2 > 0 we put 2 -c 2 \u003d b 2

Then the last equation will take the form:

is the equation of an ellipse in its canonical form.

The shape of the ellipse depends on the ratio: at b= the ellipse turns into a circle. The equation will take the form . The ratio is often used as a characteristic of an ellipse. This value is called the eccentricity of the ellipse, moreover, 0< <1 так как 0

Study of the shape of an ellipse.

1) the equation of the ellipse contains x and y, only to an even degree, so the ellipse is symmetrical about the axes Ox and Oy, as well as about t.O (0,0), which is called the center of the ellipse.

2) find the points of intersection of the ellipse with the coordinate axes. By setting y=0 we find A 1 ( ,0) and A 2 (- ,0) in which the ellipse intersects Ox. Putting x=0, we find B 1 (0,b) and B 2 (0,-b). Points A 1, A 2, B 1, B 2 are called the vertices of the ellipse. The segments A 1 A 2 and B 1 B 2 , as well as their lengths 2 and 2b, are called the major and minor axes of the ellipse, respectively. The numbers and b are, respectively, the major and minor semiaxes.

A 1 ( ,0)

A2(- ,0)

B 2 (0,b)

Therefore, all points of the ellipse lie inside the rectangle formed by the lines x=± ,y=±b. (Fig.2.)

4) In the equation of an ellipse, the sum of non-negative terms is equal to one. Therefore, as one term increases, the other will decrease, that is, if |x| increases, then |y| - decreases and vice versa. From all that has been said, it follows that the ellipse has the shape shown in Fig.2. (oval closed curve).

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Lecture number 9. Topic 3: Lines of the second order

Let a line defined by a second-degree equation be given in some DSC

where coefficients
are not equal to zero at the same time. This line is called curve or second order line.

It may happen that there are no points
with real coordinates satisfying equation (1). In this case, it is considered that equation (1) defines an imaginary line of the second order. For instance,
 This is the equation of the imaginary circle.

Let us consider three important particular cases of equation (1).

3.1. Ellipse

The ellipse is defined by the equation

(2)

Odds a and b are called the major and minor semiaxes, respectively, and equation (2) is canonical ellipse equation.

Let's put
and mark on the axis O Xpoints

called tricks ellipse. Then the ellipse can be defined as

locus of points, the sum of the distances from which to the foci is a constant value equal to 2a.

at

b

M K

— aF 1 O F 2 a x

— b

Let's show it. Let the point
 the current point of the ellipse. In this case, we obtain Then the equality

Expression (3) can be represented as

and square both sides of the expression

From here we get

Let's square this expression again and use the relation
, then

(4)

Dividing both parts of expression (4) by
, we finally obtain the canonical equation of the ellipse

Let us study equation (2). If we replace in the equation, then equation (2) will not change. This means that the ellipse is symmetrical about the coordinate axes. Therefore, let us consider in detail the part of the ellipse located in the first quarter. It is determined by the equation
It is clear that the ellipse passes through the points
. Having performed a schematic construction in the first quarter, we will display its graph symmetrically in all quarters. Thus, the ellipse is a continuous closed curve. The points are called the vertices of the ellipse.

Attitude
calledeccentricityellipse. For ellipse
.

Direct
called ellipse directrixes.

The following directrix property holds:

The ratio of the distances from the focus and the directrix for the points of the ellipse is a constant value equal to the eccentricity, i.e.

It is proved in the same way as equality (3).

Remark 1. Circle
is a special case of an ellipse. For her

3.2. Hyperbola

The canonical equation of a hyperbola has the form

those. in equation (1) we must put

Odds a and b are called the real and imaginary semiaxes, respectively.

Putting
, mark on the axis O Xpoints
called tricks hyperbole. Then the hyperbola can be defined as

locus of points, the difference in distances from which to foci in absolute value is equal to 2a, i.e.

at

TO M

F 1 —a O aF 2 X

The proof is similar to that for the ellipse. By the form of the equation of the hyperbola, we also conclude that its graph is symmetrical about the axes of the coordinate system. The part of the hyperbola lying in the first quadrant has the equation
It can be seen from this equation that for sufficiently largeXhyperbola close to a straight line
. After a schematic construction in the first quarter, we display the graph symmetrically in all quarters.

points
called peaks hyperbole. Direct
called asymptotes are the lines to which the branches of the hyperbola tend, without intersecting them.

The relation is calledeccentricityhyperbole. For hyperbole
.

Straight lines are called directors hyperbole. For the directrixes of a hyperbola, the property is similar to that for the directrixes of an ellipse.

Example. Find the equation of an ellipse whose vertices are at foci and whose foci are at the vertices of a hyperbola
.

By condition
a

Finally we get

10.3. Parabola

The parabola is defined by the canonical equation
those. in equation (1) we must put

TO coefficientR called TOat

focal setting. M

Note on the O axis Xpoint

called focus

- ellipse;

- parabola;

- hyperbole.

transcript

1 Chapter LINES OF THE SECOND ORDER ON THE PLANE.1. Ellipse, hyperbola, parabola Definition. An ellipse is the set of all points in the plane for which the sum of the distances to two given points F 1 and F is a constant value a, exceeding the distance between F 1 and. M(, x) F 1 O F x The points F 1 and F are called the foci of the ellipse, and the distance FF 1 between them is the focal length, which is denoted by c. Let the point M belong to the ellipse. The segments F1 M and F M are called the focal radii of the point M. Let F1F = c. By definition, a > c. Consider a rectangular Cartesian coordinate system Ox, in which the foci F 1 and F are located on the x-axis symmetrically with respect to the origin. In this coordinate system, the ellipse is described by the canonical equation: x + = 1, a b 1

2. where b= a c Parameters a and b are called, respectively, the major and minor semi-axes of the ellipse. The eccentricity of an ellipse is the number ε, equal to the ratio of half of its focal distance c to the semi-major axis, i.e. ε =. The eccentricity of the ellipse a satisfies the inequalities 0 ε< 1. Случай c = 0 соответствует окружности, эксцентриситет окружности равен нулю. Фокальные радиусы точки M(x,) эллипса могут быть найдены по формулам r 1 = a ε x, r = a+ ε x. Нормальное уравнение окружности имеет вид (x c) + (d) = R. Определение. Гиперболой называется множество всех точек плоскости, для которых абсолютная величина разности расстояний до данных точек F 1 и F есть величина постоянная, равная a. Точки F 1 и F называются фокусами гиперболы, а расстояние между ними фокальным расстоянием, которое обозначается c. Отрезки F1 M и F M называются фокальными радиусами точки M (x,) гиперболы. Рассмотрим прямоугольную декартову систему координат Ox, в которой фокусы F 1 и F расположены на оси абсцисс симметрично относительно начала координат. M (x,) F 1 F x Рис. 3

3 The canonical equation of a hyperbola has the form x a = b 1,. where b= c a The numbers a and b are called respectively the real and imaginary semiaxes of the hyperbola. There are no hyperbola points inside the region defined by the inequality. x a b Definition. The asymptotes of a hyperbola are straight lines b b given by the equations = x, = x. a a The focal radii of the point M(x,) of the hyperbola can be found by the formulas r 1 = ε x a, r = ε x+ a. The eccentricity of a hyperbola, as for an ellipse, is determined by the formula ε =. It is easy to check that the inequality ε a >1 is true for the eccentricity of the hyperbola. Definition. A parabola is the set of all points in the plane for which the distance to a given point F is equal to the distance to a given line d that does not pass through the point F. The point F is called the focus of the parabola, and the line d is called the directrix. The distance from the focus to the directrix is ​​called the parameter of the parabola and is denoted by p. d M (x,) F x 4 3

4 Let's choose the origin O of the Cartesian coordinate system in the middle of the segment FD, which is a perpendicular dropped from the point F to the line d. In this coordinate system, the focus F has coordinates F p p ;0, and the directrix d is given by the equation x + = 0. The canonical equation of a parabola is: = px. The parabola is symmetrical about the OF axis, called the parabola axis. The point O of intersection of this axis with the parabola is called the vertex of the parabola. Focal radius of the point M (x,) i.e. its p distance to the focus is found by the formula r = x+. 10B.. General equation of a second order line A second order line is a set of points in the plane whose coordinates x and which satisfy the equation ax + a x+ a + a x+ a + a =0, ​​11 1 where a11, a1, a, a10, a0, a00 some real numbers, and a, a, a are not equal to zero at the same time. This equation is called the general second-order curve equation and can also be written in vector form rr rr (Ax, x) + (b, x) + a = 0, where 00 a11 a1 rr A =, a1 ab = (a10; a0) , x = (x;). T Since A = A, then A is a quadratic matrix r r r f (x) = (Ax, x) = a x + a x+ a Ellipse, hyperbola and parabola are examples of second-order curves in the plane. In addition to the named curves, there are other types of curves of the second order, which are connected with x by straight lines. So, for example, equation = 0, where a 0, b 0, a b 4

5 defines a pair of intersecting lines on the plane. The coordinate systems in which the equation of the curve takes the simplest form are called canonical. Using the composition of transformations: rotation of the axes by an angle α, parallel transfer of the origin to the point (x0; 0) and reflection about the abscissa axis, the equation of the second-order curve is reduced to one of the canonical equations, the main of which were listed above. 11BExamples 1. Compose the canonical equation of an ellipse centered at the origin and foci located on the abscissa axis, if it is known that its eccentricity ε = and the point N(3;) lies on the 3rd ellipse. x a b Ellipse equation: + = 1. We have that =. a b a 3 9 Hence we calculate that a = b. Substituting the coordinates of the point N(3;) into the equation, we obtain + = 1 and then b = 9 and a b 81 a = = 16,. Therefore, the canonical equation of the ellipse is 5 x + = 1. 16, 9. Compose the canonical equation of a hyperbola with the center at the origin and foci located on the abscissa axis, if the point M 1 (5; 3) of the hyperbola and the eccentricity ε = are given. x The canonical equation of the hyperbola = 1. From the equality a b a + b = we have b = a 5 9. Hence = 1 and a =16. Therefore, the canonical equation of the ellipse = a a a x 16 5

6 3. Find points on the parabola = 10x whose focal radius is 1.5. Note that the parabola is located in the right half-plane. If M (x; lies on a parabola, then x 0. Parameter p = 5. Let (;)) M x be the desired point, F is the focus, () the directrix of the parabola. Then F,5; 0, d: x=,5. Since FM = ρ(M, d), then x +,5 = 1.5, 10 Answer: () 1 10;10 x =, = 100, =± 10. So, we got two points. M10; 10 M, () 4. On the right branch of the hyperbola given by the equation x = 1, find a point whose distance from the right focus is 16 9 two times less than its distance from the left focus. For the right branch of the hyperbola, the focal radii are defined by the formulas r 1 = ε x a and r = ε x + a. Therefore, we obtain the equation ε x + a = (ε x a). For a given hyperbola a = 4, 5 c = 5 and ε =. Therefore, x = 9.6. From here we have = ± x 16 = ± d Answer: two points M 1 (9.6; 0.6 119), (9.6; 0.6 119) M. 5. Find the equation of the line, for any point of which the distance ratio to the point F (3;0) to the distance to the straight line 1 x 8= 0 is equal to ε =. Specify the name of the line and its parameters. Mx; the desired line, the equality is true: For an arbitrary point () FM (x 3) + 1 = =. ρ(Ml,) x 8 6

7 Hence we have [(x 3) + ] = (x 8). Opening the brackets and rearranging the terms, we get (x+) + = 50, i.e. (x+) + = Answer: the desired line is an ellipse centered at a point and semi-axes a = 5 and b = Find the equation of the hyperbola Old coordinates coordinates O () x ; 0; ;, ;. C(;0) = 8 in the new system (x ;) and new ones (zt ;) are related by the matrix equality 1 1 x z 1 z+ t = 1 1 t = z t. Hence, the equation x = 8 z+ t z t = 8, zt = 4. Answer: zt = 4. γ:4x 4x+ 8x+ 4+ 3= 0 to canonical form. in new coordinates has the form Consider the quadratic form () q x, = 4x 4x+. The 4 form matrix q has eigenvalues ​​5 and 0 and the corresponding orthonormal vectors and

8 z 1 1 x. t = 5 1 Let's express old coordinates (x;) through new ones (zt) ; : 1 1 z+ tx 1 z = 1 t =, 1 zt means x = z+ t, = z+ t ) () ()() = 5z 4 5z+ 3= z 5 4 z 5 + 3= z 5 1 z 5 3. Hence, in the new coordinates, the curve γ is given by the equation 1 3 γ: zz =. Setting = z, x = t, we obtain γ: =, 1 whence we find the canonical equation of the curve γ: = 0 in canonical coordinates = 5 x 1 1 x Note that the curve γ is a pair of parallel lines. 1BAppendices to economic and financial problems 8. Let Anya, Boris and Dmitry each have 150 rubles to buy fruit. It is known that 1 kg of pears costs 15 monetary units, and 1 kg of apples costs 10 monetary units. At the same time, each of the three

9 has a utility function for which he wants to maximize his purchase. Let x1 kg of pears and x kg of apples be bought. These utility functions are as follows: u = x + x for Anya, 1 A 1 x u B = +x for Boris, and ud = x1 x for Dmitry. It is required to find a purchase plan (x1, x) for Anya, Boris and Dmitry, under which they provide the maximum of their utility function. x Fig. 5 The problem under consideration can be solved geometrically. To solve this problem, the concept of a level line should be introduced. x x 1 Fig. 6 The level line of a function z = f(x,) is the set of all points on the plane on which the function retains a constant value equal to h. x9

10 In this case, the solution will also use the initial ideas about the geometric areas on the plane, given by linear inequalities (see subsection 1.4). x x 1 Fig. 7 The level lines of the functions ua, u B and u D are straight lines, ellipses and hyperbolas for Anya, Boris and Dmitry, respectively. By the meaning of the problem, we assume that x1 0, x 0. On the other hand, the budget constraint is written as the inequality 15x1+ 10x 150. Dividing the last inequality by 10, we get 3x1+ x 30, or + 1. It is easy to see that x1 x is the solution area of ​​this inequality together with the non-negativity conditions is a triangle bounded by the lines x1 = 0, x = 0 and 3x1+ x =

11 X * X * Fig. 8 Fig. 9 Based on geometric patterns, it is now easy to establish that uamax = ua(0.15) = 15, ubmax = ub(0.15) = 5, and udmax = ud(Q). The coordinates of the point Q of the tangency of the level hyperbola of the side of the budget triangle must already be calculated analytically. To do this, note that the point Q satisfies three equations: xx 1 = h, 3x1 + x = 30, h 3 x " = =. x1 X * Fig

12 Eliminating h from the equations, we obtain the coordinates of the point Q= (x, x) = (5;7.5). 1 Answer: Q= (x1, x) = (5;7.5). 9. Nonlinear model of costs and profits of the company. Let the firm produce multi-purpose equipment of two types A and B in the amount of x and units of production, respectively. At the same time, the company's income for the year is expressed by the income function Rx (,) = 4x+, and the production costs are expressed by the cost function 1 1 Cx (,) = 7.5+ x + 4 where the company receives the maximum profit. Determine the production plan (x, ) at 3

13 The profit function is compiled as the difference between the income function and the cost function: 1 1 Π (x,) = R(x,) C(x,) = 4x+ 7.5 x. 4 Having done the transformations, we bring the last expression to the form 1 1 Π (x,) = 9 (x 8) (1). 4 The level lines for the profit function look like (x 8) (1) = h. 4 Each level line 0 h 9 is an ellipse centered at the origin. From the resulting expression it is easy to see that the maximum of the profit function is 9 and is achieved at x= 8, = 1. Answer: x = 8, = 1. 13BExercises and test questions.1. Write the normal equation for a circle. Find the coordinates of the center and the radius of the circle: a) x + + 8x 6=0; b) x x = 0... Write the equation of a circle passing through the points M 1 (1;), M (0; 1), M 3 (3; 0)..3. Define an ellipse and write its canonical equation. Write the canonical equation of an ellipse if 1 its eccentricity is equal to ε =, and the semi-major axis is equal to Compose an equation of an ellipse whose foci lie on the ordinate axis symmetrically about the origin, knowing, in addition, that the distance between its foci c = 4 and the eccentricity ε = Give determination of the eccentricity of an ellipse. Find the eccentricity of an ellipse if its major axis is four times its minor axis. 33

14.6. Define a hyperbola and write its canonical equation. A straight line is drawn through the point M (0; 0.5) and the right vertex of the hyperbola given by the equation x = 1. Find the coordinates of the second point of intersection of the line and the hyperbola Define the eccentricity of the hyperbola. Write its canonical equation if a = 1, b = 5. What is the eccentricity of this hyperbola?.8. Write the equations for the asymptotes of the hyperbola given by its canonical equation. Write the equation of the hyperbola 3 if its asymptotes are given by the equations =± x and the hyperbola 5 passes through the point M (10; 3 3)..9. Define a parabola and write its canonical equation. Write the canonical equation of a parabola if the x-axis is its axis of symmetry, its vertex lies at the origin and the length of the chord of the parabola perpendicular to the Ox axis is 8, and the distance of this chord from the vertex is On the parabola = 1x find the point whose focal radius is Sentence and the demand for some good are given by the functions p = 4q 1, p = +. Find the point of market equilibrium. 1 q Generate graphs..1. Andrei, Katya and Nikolai are going to buy oranges and bananas. Buy x1 kg of oranges and x kg of bananas. Each of the three has his own utility function, which shows how useful he considers his purchase. These utility functions are as follows: u = x + x for Andrey, 1 4 A 4 1 u K = x + x for Katya, and un = x1 x for Nikolai. a) Plot the level lines of the utility function for level values ​​h=1, 3. b) For each, arrange in order of preference to buy rrr ki r = (4.1), s = (3.8), t = (1.1 ). 34

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Lines of the second order.
Ellipse and its canonical equation. Circle

After a thorough study straight lines on the plane we continue to study the geometry of the two-dimensional world. The stakes are doubled and I invite you to visit the picturesque gallery of ellipses, hyperbolas, parabolas, which are typical representatives of second order lines. The tour has already begun, and first, a brief information about the entire exhibition on different floors of the museum:

The concept of an algebraic line and its order

A line on a plane is called algebraic, if in affine coordinate system its equation has the form , where is a polynomial consisting of terms of the form ( is a real number, are non-negative integers).

As you can see, the equation of an algebraic line does not contain sines, cosines, logarithms, and other functional beau monde. Only "x" and "y" in integer non-negative degrees.

Line order is equal to the maximum value of the terms included in it.

According to the corresponding theorem, the concept of an algebraic line, as well as its order, do not depend on the choice affine coordinate system, therefore, for the ease of being, we consider that all subsequent calculations take place in Cartesian coordinates.

General Equation the second-order line has the form , where are arbitrary real numbers (it is customary to write with a multiplier - "two"), and the coefficients are not simultaneously equal to zero.

If , then the equation simplifies to , and if the coefficients are not simultaneously equal to zero, then this is exactly general equation of a "flat" straight line, which represents first order line.

Many understood the meaning of the new terms, but, nevertheless, in order to 100% assimilate the material, we stick our fingers into the socket. To determine the line order, iterate over all terms its equations and for each of them find sum of powers incoming variables.

For instance:

the term contains "x" to the 1st degree;
the term contains "Y" to the 1st power;
there are no variables in the term, so the sum of their powers is zero.

Now let's figure out why the equation sets the line second order:

the term contains "x" in the 2nd degree;
the term has the sum of the degrees of the variables: 1 + 1 = 2;
the term contains "y" in the 2nd degree;
all other terms - lesser degree.

Maximum value: 2

If we additionally add to our equation, say, , then it will already determine third order line. It is obvious that the general form of the 3rd order line equation contains a “complete set” of terms, the sum of the degrees of variables in which is equal to three:
, where the coefficients are not simultaneously equal to zero.

In the event that one or more suitable terms are added that contain , then we will talk about 4th order lines, etc.

We will have to deal with algebraic lines of the 3rd, 4th and higher orders more than once, in particular, when getting acquainted with polar coordinate system.

However, let us return to the general equation and recall its simplest school variations. As examples, the parabola suggests itself, the equation of which can be easily reduced to general view, and a hyperbola with an equivalent equation . However, not everything is so smooth ....

A significant drawback of the general equation is that it is almost always not clear which line it defines. Even in the simplest case, you will not immediately realize that this is hyperbole. Such layouts are good only at a masquerade, therefore, in the course of analytical geometry, a typical problem is considered reduction of the 2nd order line equation to the canonical form.

What is the canonical form of an equation?

This is the generally accepted standard form of the equation, when in a matter of seconds it becomes clear what geometric object it defines. In addition, the canonical form is very convenient for solving many practical tasks. So, for example, according to the canonical equation "flat" straight, firstly, it is immediately clear that this is a straight line, and secondly, the point belonging to it and the direction vector are simply visible.

Obviously, any 1st order line represents a straight line. On the second floor, there is no longer a janitor waiting for us, but a much more diverse company of nine statues:

Classification of second order lines

Via special complex actions, any second-order line equation is reduced to one of the following forms:

( and are positive real numbers)

1) is the canonical equation of the ellipse;

2) is the canonical equation of the hyperbola;

3) is the canonical equation of the parabola;

4) – imaginary ellipse;

5) - a pair of intersecting lines;

6) - couple imaginary intersecting lines (with the only real point of intersection at the origin);

7) - a pair of parallel lines;

8) - couple imaginary parallel lines;

9) is a pair of coinciding lines.

Some readers may get the impression that the list is incomplete. For example, in paragraph number 7, the equation sets the pair direct, parallel to the axis, and the question arises: where is the equation that determines the lines parallel to the y-axis? Answer: it not considered canon. The straight lines represent the same standard case rotated by 90 degrees, and an additional entry in the classification is redundant, since it does not carry anything fundamentally new.

So there are nine and only nine various kinds lines of the 2nd order, but in practice the most common ellipse, hyperbola and parabola.

Let's look at the ellipse first. As usual, I focus on those points that have great importance for solving problems, and if you need a detailed derivation of formulas, proofs of theorems, please refer, for example, to the textbook by Bazylev / Atanasyan or Aleksandrov.

Ellipse and its canonical equation

Spelling ... please do not repeat the mistakes of some Yandex users who are interested in "how to build an ellipse", "the difference between an ellipse and an oval" and "elebs eccentricity".

The canonical equation of an ellipse has the form , where are positive real numbers, and . I will formulate the definition of an ellipse later, but for now it's time to take a break from talking and solve a common problem:

How to build an ellipse?

Yes, take it and just draw it. The assignment is common, and a significant part of the students do not quite competently cope with the drawing:

Example 1

Construct an ellipse given by the equation

Solution: first we bring the equation to the canonical form:

Why bring? One of the advantages of the canonical equation is that it allows you to instantly determine ellipse vertices, which are at the points . It is easy to see that the coordinates of each of these points satisfy the equation .

In this case :


Section called major axis ellipse;
section – minor axis;
number called semi-major axis ellipse;
number – semi-minor axis.
in our example: .

To quickly imagine what this or that ellipse looks like, just look at the values ​​\u200b\u200bof "a" and "be" of its canonical equation.

Everything is fine, neat and beautiful, but there is one caveat: I completed the drawing using the program. And you can draw with any application. However, in harsh reality, a checkered piece of paper lies on the table, and mice dance around our hands. People with artistic talent, of course, can argue, but you also have mice (albeit smaller ones). It is not in vain that mankind invented a ruler, a compass, a protractor and other simple devices for drawing.

For this reason, we are unlikely to be able to accurately draw an ellipse, knowing only the vertices. Still all right, if the ellipse is small, for example, with semiaxes. Alternatively, you can reduce the scale and, accordingly, the dimensions of the drawing. But in the general case it is highly desirable to find additional points.

There are two approaches to constructing an ellipse - geometric and algebraic. I don’t like building with a compass and ruler because of the short algorithm and the significant clutter of the drawing. In case of emergency, please refer to the textbook, but in reality it is much more rational to use the tools of algebra. From the ellipse equation on the draft, we quickly express:

The equation is then split into two functions:
– defines the upper arc of the ellipse;
– defines the lower arc of the ellipse.

The ellipse given by the canonical equation is symmetrical with respect to the coordinate axes, as well as with respect to the origin. And that's great - symmetry is almost always a harbinger of a freebie. Obviously, it is enough to deal with the 1st coordinate quarter, so we need a function . It suggests finding additional points with abscissas . We hit three SMS on the calculator:

Of course, it is also pleasant that if a serious error is made in the calculations, then this will immediately become clear during the construction.

We mark points in the drawing (red color), symmetrical points on the remaining arcs ( blue color) and neatly connect the whole company with a line:


It is better to draw the initial sketch thinly and thinly, and only then apply pressure to the pencil. The result should be quite a decent ellipse. By the way, would you like to know what this curve is?

Definition of an ellipse. Ellipse foci and ellipse eccentricity

An ellipse is a special case of an oval. The word "oval" should not be understood in the philistine sense ("the child drew an oval", etc.). This is a mathematical term with a detailed formulation. The purpose of this lesson is not to consider the theory of ovals and their various types, which are practically not given attention in the standard course of analytic geometry. And, in accordance with more current needs, we immediately go to the strict definition of an ellipse:

Ellipse- this is the set of all points of the plane, the sum of the distances to each of which from two given points, called tricks ellipse, is a constant value, numerically equal to the length of the major axis of this ellipse: .
In this case, the distance between the foci is less than this value: .

Now it will become clearer:

Imagine that the blue dot "rides" on an ellipse. So, no matter what point of the ellipse we take, the sum of the lengths of the segments will always be the same:

Let's make sure that in our example the value of the sum is really equal to eight. Mentally place the point "em" in the right vertex of the ellipse, then: , which was required to be checked.

Another way to draw an ellipse is based on the definition of an ellipse. Higher mathematics, at times, is the cause of tension and stress, so it's time to have another session of unloading. Please take a piece of paper or a large sheet of cardboard and pin it to the table with two nails. These will be tricks. Tie a green thread to the protruding nail heads and pull it all the way with a pencil. The neck of the pencil will be at some point, which belongs to the ellipse. Now begin to guide the pencil across the sheet of paper, keeping the green thread very taut. Continue the process until you return to the starting point ... excellent ... the drawing can be submitted for verification by the doctor to the teacher =)

How to find the focus of an ellipse?

In the above example, I depicted "ready" focus points, and now we will learn how to extract them from the depths of geometry.

If the ellipse is given by the canonical equation , then its foci have coordinates , where is it distance from each of the foci to the center of symmetry of the ellipse.

Calculations are easier than steamed turnips:

! With the meaning "ce" it is impossible to identify the specific coordinates of tricks! I repeat, this is DISTANCE from each focus to the center(which in the general case does not have to be located exactly at the origin).
And, therefore, the distance between the foci cannot be tied to the canonical position of the ellipse either. In other words, the ellipse can be moved to another place and the value will remain unchanged, while the foci will naturally change their coordinates. Please bear this in mind as you explore the topic further.

The eccentricity of an ellipse and its geometric meaning

The eccentricity of an ellipse is a ratio that can take values ​​within .

In our case:

Let's find out how the shape of an ellipse depends on its eccentricity. For this fix the left and right vertices of the ellipse under consideration, that is, the value of the semi-major axis will remain constant. Then the eccentricity formula will take the form: .

Let's start to approximate the value of the eccentricity to unity. This is only possible if . What does it mean? ...remembering tricks . This means that the foci of the ellipse will "disperse" along the abscissa axis to the side vertices. And, since “the green segments are not rubber”, the ellipse will inevitably begin to flatten, turning into a thinner and thinner sausage strung on an axis.

In this way, the closer the eccentricity of the ellipse is to one, the more oblong the ellipse is.

Now let's simulate the opposite process: the foci of the ellipse went towards each other, approaching the center. This means that the value of "ce" is getting smaller and, accordingly, the eccentricity tends to zero: .
In this case, the “green segments”, on the contrary, will “become crowded” and they will begin to “push” the line of the ellipse up and down.

In this way, the closer the eccentricity value is to zero, the more the ellipse looks like... look at the limiting case, when the foci are successfully reunited at the origin:

A circle is a special case of an ellipse

Indeed, in the case of equality of the semiaxes, the canonical equation of the ellipse takes the form, which reflexively transforms to the well-known circle equation from the school with the center at the origin of the radius "a".

In practice, the notation with the “speaking” letter “er” is more often used:. The radius is called the length of the segment, while each point of the circle is removed from the center by the distance of the radius.

Note that the definition of an ellipse remains completely correct: the foci matched, and the sum of the lengths of the matched segments for each point on the circle is a constant value. Since the distance between foci is the eccentricity of any circle is zero.

A circle is built easily and quickly, it is enough to arm yourself with a compass. However, sometimes it is necessary to find out the coordinates of some of its points, in this case we go the familiar way - we bring the equation to a cheerful Matan's form:

is the function of the upper semicircle;
is the function of the lower semicircle.

Then we find the desired values, differentiable, integrate and do other good things.

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Example 2

Compose the canonical equation of an ellipse if one of its foci and the semi-minor axis are known (the center is at the origin). Find vertices, additional points and draw a line on the drawing. Calculate the eccentricity.

Solution and drawing at the end of the lesson

Let's add an action:

Rotate and translate an ellipse

Let's return to the canonical equation of the ellipse, namely, to the condition, the riddle of which has been tormenting inquisitive minds since the first mention of this curve. Here we have considered an ellipse , but in practice can not the equation ? After all, here, however, it seems to be like an ellipse too!

Such an equation is rare, but it does come across. And it does define an ellipse. Let's dispel the mystic:

As a result of the construction, our native ellipse is obtained, rotated by 90 degrees. That is, - it non-canonical entry ellipse . Record!- the equation does not specify any other ellipse, since there are no points (foci) on the axis that would satisfy the definition of an ellipse.