220400 Algebra and Geometry Tolstikov A.V.

Lectures 16. Bilinear and quadratic forms.

Plan

1. Bilinear form and its properties.

2. Quadratic form. Quadratic matrix. Coordinate transformation.

3. Reduction of the quadratic form to the canonical form. Lagrange's method.

4. The law of inertia of quadratic forms.

5. Reduction of the quadratic form to the canonical form by the method of eigenvalues.

6. Silverst's criterion for positive definiteness of a quadratic form.

1. Course in Analytical Geometry and Linear Algebra. Moscow: Nauka, 1984.

2. Bugrov Ya.S., Nikolsky S.M. Elements of linear algebra and analytic geometry. 1997.

3. Voevodin V.V. Linear Algebra .. M .: Nauka 1980.

4. Collection of tasks for technical colleges. Linear algebra and the foundations of mathematical analysis. Ed. Efimova A.V., Demidovich B.P .. Moscow: Nauka, 1981.

5. Butuzov V.F., Krutitskaya N.Ch., Shishkin A.A. Linear algebra in questions and problems. Moscow: Fizmatlit, 2001.

, , , ,

1. Bilinear form and its properties. Let be V - n-dimensional vector space over the field P.

Definition 1.Bilinear formdefined on V, such a mapping is called g: V 2 ® P, which to each ordered pair ( x , y ) vectors x , y from puts to V match the number from the field Pdenoted g(x , y ), and linear in each of the variables x , y , i.e. possessing properties:

1) ("x , y , z Î V) g(x + y , z ) = g(x , z ) + g(y , z );

2) ("x , y Î V) ("a Î P) g(a x , y ) \u003d a g(x , y );

3) ("x , y , z Î V) g(x , y + z ) = g(x , y ) + g(x , z );

4) ("x , y Î V) ("a Î P) g(x , a y ) \u003d a g(x , y ).

Example 1... Any dot product defined on a vector space V is a bilinear form.

2 ... Function h(x , y ) = 2x 1 y 1 - x 2 y 2 + x 2 y 1, where x = (x 1 , x 2), y = (y 1 , y 2) Î R 2, bilinear form on R 2 .

Definition 2.Let be v = (v 1 , v 2 ,…, v n V.Bilinear matrixg(x , y ) on the basisv called the matrix B=(b ij) n ´ nwhose elements are calculated by the formula b ij = g(v i, v j):

Example 3... Bilinear matrix h(x , y ) (see example 2) with respect to the basis e 1 = (1,0), e 2 \u003d (0,1) is equal.

Theorem 1. Let beX, Y- coordinate columns respectively vectors x , y in the basisv, B - matrix of bilinear formg(x , y ) on the basisv. Then the bilinear form can be written as

g(x , y )=X t BY. (1)

Evidence. By the properties of the bilinear form, we obtain

Example 3... Bilinear form h(x , y ) (see example 2) can be written as h(x , y )=.

Theorem 2. Let be v = (v 1 , v 2 ,…, v n), u = (u 1 , u 2 ,…, u n) - two bases of vector space V, T- matrix of transition from basisv to the basisu. Let be B= (b ij) n ´ n and FROM=(with ij) n ´ n - bilinear matricesg(x , y ) respectively with respect to the basesv andu. Then

FROM= T t BT.(2)

Evidence. By the definition of the transition matrix and the bilinear matrix, we find:



Definition 2.Bilinear form g(x , y ) is called symmetrical, if a g(x , y ) = g(y , x ) for any x , y Î V.

Theorem 3. Bilinear formg(x , y )- symmetric if and only if the matrix of bilinear form is symmetric with respect to any basis.

Evidence.Let be v = (v 1 , v 2 ,…, v n) is the basis of the vector space V, B= (b ij) n ´ n - matrices of bilinear form g(x , y ) with respect to the basis v.Let the bilinear form g(x , y ) - symmetric. Then, by Definition 2, for any i, j = 1, 2,…, n we have b ij = g(v i, v j) = g(v j, v i) = b ji... Then the matrix B - symmetrical.

Conversely, let the matrix B - symmetrical. Then B t= B and for any vectors x = x 1 v 1 + …+ x n v n = vX, y = y 1 v 1 + y 2 v 2 +…+ y n v n = vY Î V , according to formula (1), we obtain (we take into account that the number is a matrix of order 1, and does not change upon transposition)

g(x , y ) = g(x , y ) t = (X t BY) t = Y t B t X = g(y , x ).

2. Quadratic form. Quadratic matrix. Coordinate transformation.

Definition 1.Quadratic form defined on V, called mapping f: V ® P, which for any vectors x of V is defined by the equality f(x ) = g(x , x ), where g(x , y ) is a symmetric bilinear form defined on V .

Property 1.In a given quadratic form f(x ) the bilinear form can be found uniquely by the formula

g(x , y ) = 1/2(f(x + y ) - f(x )- f(y )). (1)

Evidence.For any vectors x , y Î V we obtain from the properties of the bilinear form

f(x + y ) = g(x + y , x + y ) = g(x , x + y ) + g(y , x + y ) = g(x , x ) + g(x , y ) + g(y , x ) + g(y , y ) = f(x ) + 2g(x , y ) + f(y ).

Hence formula (1) follows. In the meantime, there is no need to know about it. ”

Definition 2.Matrix of quadratic formf(x ) on the basisv = (v 1 , v 2 ,…, v n) is the matrix of the corresponding symmetric bilinear form g(x , y ) with respect to the basis v.

Theorem 1. Let beX= (x 1 , x 2 ,…, x n) t- vector coordinate column x in the basisv, B - matrix of quadratic formf(x ) on the basisv. Then the quadratic formf(x )

Introduction

quadratic form canonical form equation

Initially, the theory of quadratic forms was used to study curves and surfaces defined by second-order equations containing two or three variables. Later, this theory found other applications. In particular, in the mathematical modeling of economic processes, the objective functions can contain quadratic terms. Numerous applications of quadratic forms required the construction of a general theory, when the number of variables is equal to any, and the coefficients of the quadratic form are not always real numbers.

The theory of quadratic forms was first developed by the French mathematician Lagrange, to whom many ideas in this theory belong, in particular, he introduced the important concept of a reduced form, with the help of which he proved that the number of classes of binary quadratic forms of a given discriminant is finite. Then this theory was significantly extended by Gauss, who introduced many new concepts, on the basis of which he was able to obtain proofs of difficult and profound theorems in number theory that eluded his predecessors in this field.

The aim of the work is to study the types of quadratic forms and ways to reduce quadratic forms to the canonical form.

In this work, the following tasks are posed: select the necessary literature, consider definitions and main theorems, solve a number of problems on this topic.

Reducing the quadratic form to the canonical form

The origins of the theory of quadratic forms lie in analytic geometry, namely in the theory of curves (and surfaces) of the second order. It is known that the equation of the central curve of the second order in the plane, after transferring the origin of rectangular coordinates to the center of this curve, has the form

that in the new coordinates the equation of our curve will have the "canonical" form

in this equation, the coefficient of the product of unknowns is, therefore, zero. The transformation of coordinates (2) can be interpreted, obviously, as a linear transformation of unknowns, moreover, nondegenerate, since the determinant of its coefficients is equal to one. This transformation is applied to the left side of equation (1), and therefore we can say that the left side of equation (1) by a nondegenerate linear transformation (2) turns into the left side of equation (3).

Numerous applications required the construction of a similar theory for the case when the number of unknowns instead of two is equal to any, and the coefficients are either real or any complex numbers.

Generalizing the expression on the left side of equation (1), we come to the following concept.

The quadratic form of unknowns is a sum, each term of which is either the square of one of these unknowns, or the product of two different unknowns. The quadratic form is called real or complex, depending on whether its coefficients are real or can be any complex numbers.

Assuming that similar terms have already been reduced in the quadratic form, we introduce the following notation for the coefficients of this form: the coefficient at is denoted by, and the coefficient at the product for - by (compare with (1)!).

Since, however, the coefficient of this product could also be denoted through, i.e. the notation introduced by us presupposes the equality

The term can now be written as

and the whole quadratic form - in the form of the sum of all possible terms, where and independently of each other take values \u200b\u200bfrom 1 to:

in particular, when we get the term

From the coefficients it is possible to compose, obviously, a square matrix of order; it is called the matrix of the quadratic form, and its rank is the rank of this quadratic form.

If, in particular, i.e. matrix is \u200b\u200bnon-degenerate, then the quadratic form is also called non-degenerate. In view of equality (4), the elements of the matrix A symmetric with respect to the main diagonal are equal to each other, i.e. matrix A is symmetric. Conversely, for any symmetric matrix A of order, one can indicate a well-defined quadratic form (5) in unknowns, which has the elements of the matrix A by its coefficients.

The quadratic form (5) can be written in a different form, using the multiplication of rectangular matrices. Let us first agree on the following notation: if a square or generally rectangular matrix A is given, then through will denote the matrix obtained from the matrix A by transposition. If the matrices A and B are such that their product is defined, then the equality takes place:

those. the matrix obtained by transposing the product is equal to the product of the matrices obtained by transposing the factors, moreover, taken in reverse order.

Indeed, if the product AB is defined, then it will be determined how easy it is to check, and the product: the number of matrix columns is equal to the number of matrix rows. The matrix element in its th row and m column is located in the ith row and m column in the AB matrix. Therefore, it is equal to the sum of the products of the corresponding elements of the ith row of matrix A and the th column of matrix B, i.e. is equal to the sum of the products of the corresponding elements of the th column of the matrix and the th row of the matrix. This proves equality (6).

Note that the matrix A will be symmetric if and only if it coincides with its transpose, i.e. if a

Let us now denote by a column made up of unknowns.

is a matrix that has rows and one column. Transposing this matrix, we get the matrix

Composed of one line.

The quadratic form (5) with a matrix can now be written as the following product:

Indeed, the product will be a one-column matrix:

Multiplying this matrix on the left by the matrix, we get a "matrix" consisting of one row and one column, namely the right side of equality (5).

What happens to the quadratic form if the unknowns included in it are subjected to a linear transformation

Hence, by (6)

Substituting (9) and (10) into the record (7) of the form, we obtain:

The matrix В will be symmetric, since in view of equality (6), which is valid, obviously, for any number of factors, and equality equivalent to the symmetry of the matrix, we have:

Thus, the following theorem has been proved:

A quadratic form of unknowns that has a matrix, after performing a linear transformation of the unknowns with a matrix, turns into a quadratic form of new unknowns, and the product serves as the matrix of this form.

Suppose now that we are performing a non-degenerate linear transformation, i.e. , and therefore also are non-degenerate matrices. The product is obtained in this case by multiplying the matrix by non-degenerate matrices, and therefore, the rank of this product is equal to the rank of the matrix. Thus, the rank of the quadratic form does not change when performing a nondegenerate linear transformation.

Let us now consider, by analogy with the geometric problem of reducing the equation of the second-order central curve to the canonical form (3), indicated at the beginning of this section, the question of reducing an arbitrary quadratic form by some nondegenerate linear transformation to the form of the sum of squares of unknowns, i.e. to such a form when all the coefficients of the products of different unknowns are equal to zero; this special kind of quadratic form is called canonical. First, assume that the quadratic form in unknowns has already been reduced by a nondegenerate linear transformation to the canonical form

where are new unknowns. Some of the ratios may. Of course, be zeros. Let us prove that the number of nonzero coefficients in (11) is certainly equal to the rank of the form.

Indeed, since we arrived at (11) using a non-degenerate transformation, the quadratic form on the right-hand side of equality (11) must also be of rank.

However, the matrix of this quadratic form has a diagonal form

and the requirement that this matrix has a rank is equivalent to the assumption that there are exactly nonzero elements on its main diagonal.

Let us proceed to the proof of the following main theorem on quadratic forms.

Any quadratic form can be reduced by some nondegenerate linear transformation to the canonical form. If, in this case, a real quadratic form is considered, then all the coefficients of the indicated linear transformation can be considered real.

This theorem is true for the case of quadratic forms in one unknown, since any such form has a form that is canonical. We can, therefore, carry out the proof by induction on the number of unknowns, i.e. prove the theorem for quadratic forms in n unknowns, considering it already proven for forms with fewer unknowns.

The quadratic form is empty

from n unknowns. We will try to find a non-degenerate linear transformation that would single out one of the unknowns from the square, i.e. would lead to the form of the sum of this square and some quadratic form of the remaining unknowns. This goal is easily achieved if, among the coefficients in the matrix, the forms on the main diagonal are nonzero, i.e. if (12) contains the square of at least one of the unknowns with nonzero coefficients

Let, for example,. Then, as it is easy to check, the expression, which is a quadratic form, contains the same terms with the unknown as our form, and therefore the difference

will be a quadratic form containing only unknowns but not. From here

If we introduce the notation

we get

where will now be the quadratic form of the unknowns. Expression (14) is the desired expression for the form, since it is obtained from (12) by a nondegenerate linear transformation, namely, the transformation inverse to linear transformation (13), which has its determinant and therefore is not degenerate.

If there are equalities, then first you need to perform an auxiliary linear transformation, leading to the appearance of squares of unknowns in our form. Since among the coefficients in the record (12) of this form should be nonzero, otherwise there would be nothing to prove, then let, for example, i.e. is the sum of a member and members, each of which includes at least one of the unknowns.

Let us now perform a linear transformation

It will be nondegenerate, since it has the determinant

As a result of this transformation, the member of our form will take the form

those. in the form, with nonzero coefficients, the squares of two unknowns at once will appear, and they cannot cancel with any of the other terms, since each of these latter includes at least one of the unknowns; now we are in the conditions of the case already considered above, those. by another nondegenerate linear transformation, we can reduce the form to the form (14).

To complete the proof, it remains to note that the quadratic form depends on less than the number of unknowns and therefore, by the induction hypothesis, is reduced to canonical form by some nondegenerate transformation of the unknowns. This transformation, considered as a (non-degenerate, as it is easy to see) transformation of all unknowns, in which it remains unchanged, leads, therefore, (14) to the canonical form. Thus, the quadratic form by two or three non-degenerate linear transformations, which can be replaced by one non-degenerate transformation - their product, is reduced to the form of the sum of the squares of the unknowns with some coefficients. The number of these squares is equal, as we know, to the rank of the form. If, moreover, the quadratic form is real, then the coefficients both in the canonical form of the form and in the linear transformation leading to this form will be real; in fact, both the inverse linear transformation (13) and the linear transformation (15) have real coefficients.

The proof of the main theorem is complete. The method used in this proof can be applied in specific examples to actually reduce the quadratic form to the canonical form. It is only necessary, instead of the induction, which we used in the proof, to sequentially separate the squares of the unknowns using the above method.

Example 1. Reduce to canonical form a quadratic form

Since there are no squares of unknowns in this form, we first perform a nondegenerate linear transformation

with matrix

after which we get:

Now the coefficients at are nonzero, and therefore from our form we can select the square of one unknown. Assuming

those. performing a linear transformation for which the inverse will have the matrix

we will bring to mind

So far, only the square of the unknown has stood out, since the form still contains the product of two other unknowns. Using the inequality of the coefficient at to zero, we apply the above method once again. Making a linear transformation

for which the inverse has a matrix

we will finally bring the form to the canonical form

A linear transformation that brings (16) directly to the form (17) will have as its matrix the product

It is also possible to verify by direct substitution that the nondegenerate (since the determinant is equal) linear transformation

turns (16) into (17).

The theory of reduction of a quadratic form to the canonical form is constructed by analogy with the geometric theory of central curves of the second order, but cannot be considered a generalization of this latter theory. Indeed, in our theory, it is allowed to use any non-degenerate linear transformations, while the reduction of a second-order curve to the canonical form is achieved by using linear transformations of a very special form,

which are the rotation of the plane. This geometric theory can, however, be generalized to the case of quadratic forms in unknowns with real coefficients. The presentation of this generalization, called reduction of quadratic forms to principal axes, will be given below.

Reducing the quadratic form to the canonical form.

Canonical and normal quadratic form.

Linear transformations of variables.

The concept of a quadratic form.

Quadratic forms.

Definition: A homogeneous polynomial of the second degree with respect to these variables is called a quadratic form in variables.

Variables can be viewed as the affine coordinates of a point in the arithmetic space A n or as the coordinates of the vector of the n-dimensional space V n. We will denote a quadratic form in variables as.

Example 1:

If similar terms have already been reduced in quadratic form, then the coefficients at are denoted, and at () -. Thus, it is believed that. The quadratic form can be written as follows:

Example 2:

System matrix (1):

- called matrix of quadratic form.

Example: The matrices of quadratic forms in Example 1 are:

The quadratic matrix of example 2:

Linear transformation of variables is called such a transition from a system of variables to a system of variables in which old variables are expressed through new ones using the forms:

where the coefficients form a non-degenerate matrix.

If variables are considered as coordinates of a vector in Euclidean space with respect to some basis, then linear transformation (2) can be considered as a transition in this space to a new basis, with respect to which the same vector has coordinates.

In what follows, we will consider quadratic forms only with real coefficients. We will assume that the variables take only real values. If the variables in the quadratic form (1) are subjected to a linear transformation (2), then we get a quadratic form in the new variables. In what follows, we will show that with an appropriate choice of transformation (2), the quadratic form (1) can be reduced to a form containing only the squares of the new variables, i.e. ... This kind of quadratic form is called canonical... In this case, a quadratic matrix is \u200b\u200bdiagonal:.

If all coefficients can take only one of the values: -1,0,1 the corresponding form is called normal.

Example: Equation of the central curve of the second order by transition to a new coordinate system

can be reduced to the form:, and the quadratic form in this case will take the form:

Lemma 1: If the quadratic form(1) does not contain squares of variables, then using a linear transformation it can be reduced to a form containing the square of at least one variable.

Evidence: By condition, the quadratic form contains only terms with products of variables. Let it be nonzero for any different values \u200b\u200bof i and j, i.e. - one of such terms included in the quadratic form. If you perform a linear transformation, and do not change all the others, i.e. (the determinant of this transformation is nonzero), then even two terms with squares of variables will appear in quadratic form:. These terms cannot disappear when such terms are reduced, since each of the remaining terms contains at least one variable other than either from or from.

Example:

Lemma 2: If square shape (1) contains the summand with the square of the variable, for example, at least one more term with variable , then using a linear transformation, f can be converted to form from variables , having the form: (2), where g - quadratic form containing no variable .

Evidence: Let us single out in the quadratic form (1) the sum of terms containing: (3) here, g 1 denotes the sum of all terms that do not contain.

We denote

(4), where denotes the sum of all terms that do not contain.

We divide both sides of (4) by and subtract the resulting equality from (3), after reducing similar ones we will have:

The expression on the right side does not contain a variable and is a quadratic form in variables. Let's denote this expression through g, and the coefficient through, and then f will be equal to:. If we perform a linear transformation:, the determinant of which is nonzero, then g will be a quadratic form in variables, and the quadratic form f will be reduced to the form (2). The lemma is proved.

Theorem: Any quadratic form can be converted to canonical form using variable transformations.

Evidence: We use induction on the number of variables. The quadratic form of is:, which is already canonical. Suppose the theorem is true for a quadratic form in n-1 variables and prove that it is true for a quadratic form in n variables.

If f does not contain squares of variables, then by Lemma 1 it can be reduced to the form containing the square of at least one variable; by Lemma 2, the resulting quadratic form can be represented in the form (2). Because the quadratic form is dependent on n-1 variables, then according to the inductive assumption it can be reduced to the canonical form using a linear transformation of these variables to variables, if we add a formula to the formulas of this transition, then we get the formulas for the linear transformation, which leads to the canonical the form of the quadratic form contained in equality (2). The composition of all considered transformations of variables is the desired linear transformation, leading to the canonical form of the quadratic form (1).

If the quadratic form (1) contains a square of some variable, then Lemma 1 does not need to be applied. The above method is called the Lagrange method.

From the canonical form, where, you can go to the normal form, where, if, and, if, using the transformation:

Example: Bring the quadratic form to the canonical form by the Lagrange method:

Because the quadratic form f already contains the squares of some variables, then Lemma 1 does not need to be applied.

Select members containing:

3. To obtain a linear transformation that directly reduces the form f to the form (4), we first find the transformations inverse to the transformations (2) and (3).

Now, using these transformations, let's build their composition:

If we substitute the obtained values \u200b\u200b(5) into (1), we immediately obtain a representation of the quadratic form in the form (4).

From canonical form (4) using transformation

you can go to normal view:

A linear transformation that brings the quadratic form (1) to a normal form is expressed by the formulas:

Bibliography:

1. Voevodin V.V. Linear algebra. Saint Petersburg: Lan, 2008, 416 p.

2. Beklemishev DV Course of analytic geometry and linear algebra. Moscow: Fizmatlit, 2006, 304 p.

3. Kostrikin A.I. Introduction to algebra. part II. Fundamentals of algebra: a textbook for universities, -M. : Physical and mathematical literature, 2000, 368 p.

Lecture number 26 (II semester)

Topic: The law of inertia. Positive definite forms.

Definition 10.4.Canonical view quadratic form (10.1) is called the following form:. (10.4)

Let us show that in a basis of eigenvectors, the quadratic form (10.1) takes the canonical form. Let be

- normalized eigenvectors corresponding to the eigenvalues λ 1, λ 2, λ 3 matrices (10.3) in the orthonormal basis. Then the matrix of the transition from the old basis to the new one will be the matrix

... In the new basis, the matrix AND takes the diagonal form (9.7) (by the property of eigenvectors). Thus, transforming the coordinates using the formulas:

,

we obtain in the new basis the canonical form of the quadratic form with coefficients equal to the eigenvalues λ 1, λ 2, λ 3:

Remark 1. From a geometric point of view, the considered transformation of coordinates is a rotation of the coordinate system, which aligns the old coordinate axes with the new ones.

Remark 2. If some eigenvalues \u200b\u200bof matrix (10.3) coincide, one can add a unit vector orthogonal to each of them to the corresponding orthonormal eigenvectors, and thus construct a basis in which the quadratic form takes the canonical form.

Let us reduce the quadratic form to canonical form

x² + 5 y² + z² + 2 xy + 6xz + 2yz.

Its matrix has the form In the example considered in Lecture 9, the eigenvalues \u200b\u200band orthonormalized eigenvectors of this matrix were found:

Let's compose the transition matrix to the basis from these vectors:

(the order of the vectors is changed so that they form the right triple). We transform the coordinates using the formulas:

.

So, the quadratic form is reduced to the canonical form with the coefficients equal to the eigenvalues \u200b\u200bof the matrix of the quadratic form.

Lecture 11.

Curves of the second order. Ellipse, hyperbola and parabola, their properties and canonical equations. Reduction of the second order equation to the canonical form.

Definition 11.1.Curves of the second order on a plane are the lines of intersection of a circular cone with planes that do not pass through its vertex.

If such a plane intersects all generatrices of one cavity of the cone, then in the section it turns out ellipse, at the intersection of the generatrices of both cavities - hyperbola, and if the secant plane is parallel to some generatrix, then the section of the cone is parabola.

Comment. All curves of the second order are given by equations of the second degree in two variables.

Ellipse.

Definition 11.2.Ellipse is called the set of points of the plane for which the sum of the distances to two fixed points F 1 and F tricks, there is a constant value.

Comment. When points coincide F 1 and F 2 the ellipse turns into a circle.

We derive the equation of the ellipse, choosing the Cartesian system

y M (x, y) coordinates so that the axis Ohcoincided with a straight line F 1 F 2, start

r 1 r 2 coordinates - with the middle of the segment F 1 F 2. Let the length of this

segment is equal to 2 from, then in the selected coordinate system

F 1 O F 2 x F 1 (-c, 0), F 2 (c, 0). Let the point M (x, y) lies on the ellipse, and

the sum of the distances from it to F 1 and F 2 equals 2 and.

Then r 1 + r 2 = 2abut,

therefore, introducing the notation b² = a²- c² and carrying out simple algebraic transformations, we obtain canonical ellipse equation: (11.1)

Definition 11.3.Eccentricity ellipse is called the value e \u003d s / a (11.2)

Definition 11.4.Headmistress D iellipse corresponding to focus F i F i about the axis OUperpendicular to axis Ohon distance a / e from the origin.

Comment. With a different choice of the coordinate system, the ellipse can be specified not by the canonical equation (11.1), but by a second-degree equation of a different kind.

Ellipse properties:

1) An ellipse has two mutually perpendicular axes of symmetry (principal axes of the ellipse) and a center of symmetry (center of the ellipse). If an ellipse is given by a canonical equation, then its principal axes are the coordinate axes, and its center is the origin. Since the lengths of the segments formed by the intersection of the ellipse with the principal axes are 2 andand 2 b (2a>2b), then the main axis passing through the foci is called the major axis of the ellipse, and the second major axis is called the minor axis.

2) The whole ellipse is contained within the rectangle

3) Eccentricity of the ellipse e< 1.

Really,

4) The directrix of the ellipse is located outside the ellipse (since the distance from the center of the ellipse to the directrix is a / e, and e<1, следовательно, a / e\u003e a, and the whole ellipse lies in a rectangle)

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

Evidence.

Distances from point M (x, y) before the focuses of the ellipse can be represented as follows:

Let's compose the directrix equations:

(D 1), (D 2). Then From here r i / d i \u003d e, as required to prove.

Hyperbola.

Definition 11.5.Hyperbole is the set of points of the plane for which the modulus of the difference between the distances to two fixed points F 1 and F 2 of this plane, called tricks, there is a constant value.

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

|r 1 - r 2 | \u003d2a, whence If we denote b² = c² - a², from here you can get

- canonical hyperbola equation. (11.3)

Definition 11.6.Eccentricity hyperbole is called the value e \u003d s / a.

Definition 11.7.Headmistress D i focus hyperbole F i, is called a straight line located in one half-plane with F iabout the axis OUperpendicular to axis Oh on distance a / e from the origin.

Hyperbola properties:

1) The hyperbola has two axes of symmetry (the main axes of the hyperbola) and the center of symmetry (the center of the hyperbola). Moreover, one of these axes intersects the hyperbola at two points, called the vertices of the hyperbola. It is called the real axis of the hyperbola (axis Ohfor the canonical choice of the coordinate system). The other axis has no points in common with the hyperbola and is called its imaginary axis (in canonical coordinates, the axis OU). On both sides of it are the right and left branches of the hyperbola. The foci of the hyperbola are located on its real axis.

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

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

for which the real and imaginary axes are interchanged while maintaining the same asymptotes.

4) Eccentricity of hyperbola e> 1.

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

The proof can be carried out in the same way as for the ellipse.

Parabola.

Definition 11.8.Parabola is called the set of points on the plane for which the distance to some fixed point Fthis plane is equal to the distance to some fixed straight line. Point F called focus parabolas, and the straight line is its headmistress.

To derive the parabola equation, we choose the Cartesian

coordinate system so that its origin is the middle

D M (x, y) perpendicular FDout of focus on direct

r cy, and the coordinate axes were located parallel and

perpendicular to the directrix. Let the length of the segment FD

D O F x equals r... Then from the equality r \u003d d follows that

insofar as

By algebraic transformations, this equation can be reduced to the form: y² \u003d 2 px, (11.4)

called the canonical parabola equation... The quantity rcalled parameterparabolas.

Parabola properties:

1) The parabola has an axis of symmetry (parabola axis). The point of intersection of the parabola with the axis is called the apex of the parabola. If a parabola is given by a canonical equation, then its axis is the axis Oh,and the vertex is the origin.

2) The whole parabola is located in the right half-plane of the plane Ooh.

Comment. Using the properties of directrix ellipse and hyperbola and the definition of a parabola, we can prove the following statement:

The set of points in the plane for which the ratio ethe distance to some fixed point to the distance to some straight line is a constant value, it is an ellipse (for e<1), гиперболу (при e\u003e 1) or a parabola (for e=1).

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