The canonical equation of an ellipse has the form

where a is the semi-major axis; b - minor semiaxis. The points F1(c,0) and F2(-c,0) − c are called

a, b - semiaxes of the ellipse.

Finding foci, eccentricity, directrix of an ellipse if its canonical equation is known.

Definition of a hyperbola. Foci of hyperbole.

Definition. A hyperbola is a set of points in a plane for which the modulus of the difference in distances from two given points, called foci, is a constant value, less than the distance between the foci.

By definition, |r1 – r2|= 2a. F1, F2 are the foci of the hyperbola. F1F2 = 2c.

The canonical equation of a hyperbola. Semiaxes of a hyperbola. Construction of a hyperbola if its canonical equation is known.

Canonical equation:

The semi-major axis of the hyperbola is half the minimum distance between the two branches of the hyperbola, on the positive and negative sides of the axis (left and right relative to the origin). For a branch located on the positive side, the semi-axis will be equal to:

If we express it in terms of the conic section and the eccentricity, then the expression will take the form:

Finding foci, eccentricity, directrix of a hyperbola if its canonical equation is known.

Eccentricity of a hyperbola

Definition. The ratio is called the eccentricity of the hyperbola, where c -

half the distance between the foci, and is the real semiaxis.

Taking into account the fact that c2 - a2 = b2:

If a \u003d b, e \u003d, then the hyperbola is called equilateral (equilateral).

Directrixes of hyperbole

Definition. Two lines perpendicular to the real axis of the hyperbola and located symmetrically about the center at a distance a / e from it are called the directrixes of the hyperbola. Their equations are:

Theorem. If r is the distance from an arbitrary point M of the hyperbola to some focus, d is the distance from the same point to the directrix corresponding to this focus, then the ratio r/d is a constant value equal to the eccentricity.

Definition of a parabola. Focus and directrix of a parabola.

Parabola. A parabola is the locus of points, each of which is equally distant from a given fixed point and from a given fixed line. The point referred to in the definition is called the focus of the parabola, and the straight line is called its directrix.

The canonical equation of a parabola. parabola parameter. Construction of a parabola.

The canonical equation of a parabola in a rectangular coordinate system is: (or if the axes are reversed).

The construction of a parabola for a given value of the parameter p is performed in the following sequence:

Draw the axis of symmetry of the parabola and lay on it the segment KF=p;

Directrix DD1 is drawn through the point K perpendicular to the axis of symmetry;

The segment KF is divided in half to get the vertex 0 of the parabola;

A number of arbitrary points 1, 2, 3, 5, 6 are measured from the top with a gradually increasing distance between them;

Through these points, auxiliary lines are drawn perpendicular to the axis of the parabola;

On auxiliary straight lines, serifs are made with a radius equal to the distance from the straight line to the directrix;

The resulting points are connected by a smooth curve.

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 example:

the term contains "x" to the 1st degree;
the term contains "Y" to the 1st degree;
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. Examples are the parabola, whose equation can be easily reduced to a general form, and the 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 problems. 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

With the help of a special set of actions, any second-order line equation is reduced to one of the following types:

( 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 the additional entry in the classification is redundant, since it does not carry anything fundamentally new.

Thus, there are nine and only nine different types of 2nd order lines, but in practice the most common are ellipse, hyperbola and parabola.

Let's look at the ellipse first. As usual, I focus on those points that are of 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 :


Line segment called major axis ellipse;
line segment – 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.

Mark points on the drawing (red color), symmetrical points on the other arcs (blue color) and carefully 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.

The article, of course, is for reference only, but how can one live without love in the world? Creative task for independent solution

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, - this is 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.

Theorem. In the canonical coordinate system for an ellipse, the ellipse equation has the form:

Proof. We will carry out the proof in two stages. At the first stage, we will prove that the coordinates of any point lying on the ellipse satisfy equation (4). At the second stage, we will prove that any solution to equation (4) gives the coordinates of a point lying on an ellipse. From here it will follow that equation (4) is satisfied by those and only those points of the coordinate plane that lie on the ellipse. From here and from the definition of the equation of the curve, it will follow that equation (4) is the equation of an ellipse.

1) Let the point M(x, y) be a point of the ellipse, i.e. the sum of its focal radii is 2a:

We use the formula for the distance between two points on the coordinate plane and find the focal radii of a given point M using this formula:

Where do we get:

Let's move one root to the right side of the equality and square it:

Reducing, we get:

We give similar ones, reduce by 4 and isolate the radical:

.

We square

Open the brackets and abbreviate to:

from where we get:

Using equality (2), we obtain:

.

Dividing the last equality by , we obtain equality (4), etc.

2) Now let a pair of numbers (x, y) satisfy equation (4) and let M(x, y) be the corresponding point on the Oxy coordinate plane.

Then from (4) it follows:

We substitute this equality into the expression for the focal radii of the point M:

.

Here we have used equality (2) and (3).

In this way, . Likewise, .

Now note that it follows from equality (4) that

Or, etc. , then the following inequality follows:

From this, in turn, it follows that

It follows from equalities (5) that , i.e. the point M(x, y) is a point of the ellipse, etc.

The theorem has been proven.

Definition. Equation (4) is called the canonical equation of the ellipse.

Definition. The canonical coordinate axes for the ellipse are called the principal axes of the ellipse.

Definition. The origin of the canonical coordinate system for an ellipse is called the center of the ellipse.

Ellipse called the locus of points of the plane, for each of which the sum of the distances to two given points of the same plane, called the foci of the ellipse, is a constant value. For an ellipse, several more equivalent definitions can be given. Those who wish can get acquainted with them in more serious textbooks on analytic geometry. Here we only note that an ellipse is a curve obtained as a projection onto a plane of a circle lying in a plane that forms an acute angle with the plane. Unlike a circle, it is impossible to write down the equation of an ellipse in a "convenient" form in an arbitrary coordinate system. Therefore, for a fixed ellipse, it is necessary to select a coordinate system so that its equation is fairly simple. Let and be the foci of the ellipse. The origin of the coordinate system is located in the middle of the segment . We direct the axis along this segment, the axis is perpendicular to this segment

24)Hyperbola

It is known from the school mathematics course that the curve defined by the equation , where is a number, is called a hyperbola. However, this is a special case of a hyperbola (an equilateral hyperbola). Definition 12 . 5 A hyperbola is a locus of points in a plane, for each of which the absolute value of the difference in distances to two fixed points of the same plane, called the foci of the hyperbola, is a constant value. Just as in the case of an ellipse, to obtain the equation of a hyperbola, we choose an appropriate coordinate system. We place the origin of coordinates in the middle of the segment between the foci, direct the axis along this segment, and the ordinate axis perpendicular to it. Theorem 12. 3 Let the distance between the foci and the hyperbola be , and the absolute value of the difference in the distances from the point of the hyperbola to the foci is . Then the hyperbola in the coordinate system chosen above has the equation (12.8) where (12.9) Proof. Let be the current point of the hyperbola (Fig. 12.9). Rice. 12 . 9 . Since the difference between two sides of a triangle is less than the third side, then , that is , . By virtue of the last inequality, the real number defined by formula (12.9) exists. By convention, the foci are , . According to the formula (10.4) for the case of a plane, we obtain By the definition of a hyperbola We write this equation in the form Both parts are squared: After bringing like terms and dividing by 4, we arrive at the equality Again, we square both parts: Expanding the bracket and bringing like terms, we obtain Taking into account formula (12.9), the equation takes the form Divide both sides of the equation by and obtain equation (12.8) Equation (12.8) is called the canonical equation of the hyperbola. Proposal 12 . 3 A hyperbola has two mutually perpendicular axes of symmetry, one of which contains the foci of the hyperbola, and a center of symmetry. If a hyperbola is given by a canonical equation, then its symmetry axes are

coordinate axes and , and the origin of coordinates is the center of symmetry of the hyperbola. Proof. It is carried out similarly to the proof of Proposition 12.1. Let us construct the hyperbola given by equation (12.8). Note that, due to symmetry, it is sufficient to plot the curve only in the first coordinate angle. We express from the canonical equation as a function, provided that , and plot this function. The domain of definition is the interval , , the function grows monotonically. Derivative exists in the entire domain of definition, except for the point . Therefore, the graph is a smooth curve (no corners). Second derivative at all points of the interval is negative, therefore, the graph is convex upwards. Let's check the graph for the presence of an asymptote for . Let the equation have an asymptote. Then, according to the rules of mathematical analysis Multiply the expression under the limit sign and divide by .

So, the graph of the function has an asymptote . It follows from the symmetry of the hyperbola that -- is also an asymptote. It remains unclear the nature of the curve in the neighborhood of the point , namely, whether the graph forms and the part of the hyperbola symmetrical about the axis at this point, the angle or hyperbola at this point is a smooth curve (there is a tangent). To solve this issue, we express from equation (12.8) through: Obviously, this function has a derivative at the point , , and the hyperbola has a vertical tangent at the point. Based on the data obtained, we draw a graph of the function (Fig. 12.10). Rice. 12 . 10.Function graph Finally, using the symmetry of the hyperbola, we obtain the curve of Figure 12.11. Rice. 12 . 11 .Hyperbole Definition 12 . 6 The intersection points of the hyperbola given by the canonical equation (12.8) with the axis are called the vertices of the hyperbola, the segment between them is called the real axis of the hyperbola. The segment of the y-axis between the points is called the imaginary axis. The numbers and are called respectively the real and imaginary semiaxes of the hyperbola. The origin of coordinates is called its center. The quantity is called the eccentricity of the hyperbola. Remark 12. 3 From equality (12.9) it follows that , that is, for the hyperbola . Eccentricity characterizes the angle between the asymptotes, the closer to 1, the smaller this angle. Remark 12. 4 In contrast to the ellipse, in the canonical equation of the hyperbola, the relation between the quantities and can be arbitrary. In particular, when we get an equilateral hyperbola, known from the school mathematics course. Her equation has a familiar form if we take , and direct the axes along the bisectors of the fourth and first coordinate angles (Fig. 12.12). Rice. 12 . 12. Equilateral hyperbola To reflect the qualitative characteristics of a hyperbola in the figure, it is enough to determine its vertices, draw asymptotes and draw a smooth curve passing through the vertices, approaching the asymptotes and similar to the curve in Figure 12.10. Example 12 . 4 Construct a hyperbola, find its foci and eccentricity. Solution. Divide both sides of the equation by 4. We get the canonical equation , . We draw asymptotes and build a hyperbola (Fig. 12.13). Rice. 12 . 13 .Hyperbola From formula (12.9) we obtain . Then tricks are , , . Example 12 . 5 Construct a hyperbola . Find its foci and eccentricity. Solution. We transform the equation to the form This equation is not a canonical equation of a hyperbola, since the signs in front of and are opposite to the signs in the canonical equation. However, if we rename the variables , , then in the new variables we get the canonical equation. The real axis of this hyperbola lies on the axis, that is, on the axis of the original coordinate system, the asymptotes have the equation, that is, the equation in the original coordinates. The real semiaxis is 5, the imaginary one is 2. In accordance with these data, we build (Fig. 12.14). Rice. 12 . 14. Hyperbola with equation From formula (12.9) we obtain , , foci lie on the real axis - , , where the coordinates are given in the original coordinate system.

Parabola

In the school course of mathematics, the parabola was studied in sufficient detail, which, by definition, was the graph of a square trinomial. Here we give another (geometric) definition of a parabola. Definition 12 . 7 A parabola is the locus of points in a plane, for each of which the distance to a fixed point of this plane, called the focus, is equal to the distance to a fixed straight line lying in the same plane and called the directrix of the parabola. To obtain the equation of a curve corresponding to this definition, we introduce a suitable coordinate system. To do this, let us drop the perpendicular from the focus to the directrix. The origin of coordinates will be located in the middle of the segment , and the axis will be directed along the segment so that its direction coincides with the direction of the vector . Draw the axis perpendicular to the axis (Fig. 12.15). Rice. 12 . fifteen . Theorem 12. 4 Let the distance between the focus and the directrix of the parabola be . Then in the chosen coordinate system the parabola has the equation (12.10) Proof. In the chosen coordinate system, the focus of the parabola is the point, and the directrix has an equation (Fig. 12.15). Let be the current point of the parabola. Then by formula (10.4) for the plane case we find The distance from the point to the directrix is ​​the length of the perpendicular dropped to the directrix from the point. From Figure 12.15 it is clear that . Then by the definition of a parabola, that is Let's square both sides of the last equation: where After reducing similar terms, we obtain equation (12.10). Equation (12.10) is called the canonical parabola equation. Proposal 12 . 4 A parabola has an axis of symmetry. If the parabola is given by the canonical equation, then the axis of symmetry coincides with the axis. Proof. Carried out in the same way as the proof (Proposition 12.1). The point of intersection of the axis of symmetry with the parabola is called the vertex of the parabola. If we rename the variables , , then the equation (12.10) can be written in a form that coincides with the usual parabola equation in a school mathematics course. Therefore, we draw a parabola without additional research (Fig. 12.16). Rice. 12 . 16. Parabola Example 12. 6 Construct a parabola . Find her focus and headmistress. Solution. The equation is the canonical equation of the parabola, , . The axis of the parabola is the axis, the vertex is at the origin, the branches of the parabola are directed along the axis. To construct, we find several points of the parabola. To do this, we assign values ​​to the variable and find the values. Take the points , , . Given the symmetry about the axis, draw a curve (Fig. 12.17) Rice. 12 . 17. The parabola given by the equation Focus lies on the axis at a distance from the top, that is, it has coordinates. The directrix has an equation , that is . The parabola, like the ellipse, has the property of reflecting light (Fig. 12.18). We state the property again without proof. Proposal 12 . 5 Let be the focus of the parabola, be an arbitrary point of the parabola, be a ray with origin at a point parallel to the axis of the parabola. Then the normal to the parabola at the point bisects the angle formed by the segment and the ray. Rice. 12 . 18. Reflection of a light beam from a parabola This property means that a beam of light that has gone out of focus, reflected from the parabola, will continue to go parallel to the axis of this parabola. Conversely, all rays coming from infinity and parallel to the axis of the parabola will converge at its focus. This property is widely used in engineering. In spotlights, a mirror is usually placed, the surface of which is obtained by rotating a parabola around its axis of symmetry (parabolic mirror). The light source in the spotlights is placed at the focus of the parabola. As a result, the spotlight gives a beam of almost parallel beams of light. The same property is used in receiving antennas for space communications and in telescope mirrors, which collect a stream of parallel beams of radio waves or a stream of parallel beams of light and concentrate it at the focus of the mirror.

26) Matrix definition. A matrix is ​​a rectangular table of numbers containing a certain number of m rows and a certain number of n columns.

Basic concepts of a matrix: The numbers m and n are called the orders of the matrix. If m=n, the matrix is ​​called square, and the number m=n is its order.

In what follows, the following notation will be used to write the matrix:

Although sometimes in the literature there is a designation:

However, for a short designation of the matrix, one capital letter of the Latin alphabet is often used, (for example, A), or the symbol ||a ij ||, and sometimes with an explanation: A=||a ij ||=(a ij) (i =1,2,...,m;j=1,2,...n)

The numbers a ij , which are part of this matrix, are called its elements. In the record a ij, the first index i means the row number, and the second index j is the column number.

For example, matrix

is a 2×3 matrix, its elements a 11 =1, a 12 =x, a 13 =3, a 21 =-2y, ...

So, we have introduced the definition of a matrix. Consider the types of matrices and give the definitions corresponding to them.

Types of matrices

Let us introduce the concept of matrices: square, diagonal, identity and zero.

Definition of a square matrix: Square matrix The nth order is called an n × n matrix.

In the case of a square matrix

the concepts of main and secondary diagonals are introduced. main diagonal matrix is ​​called the diagonal going from the upper left corner of the matrix to the lower right corner.

side diagonal of the same matrix is ​​called the diagonal going from the lower left corner to the upper right corner.

The concept of a diagonal matrix: Diagonal is a square matrix in which all elements outside the main diagonal are equal to zero.

The concept of the identity matrix: Solitary(denoted E sometimes I) is called a diagonal matrix with ones on the main diagonal.

The concept of a zero matrix: Null is a matrix all of whose elements are equal to zero.

Two matrices A and B are called equal (A=B) if they are of the same size (i.e. have the same number of rows and the same number of columns and their corresponding elements are equal). So if

then A \u003d B, if a 11 \u003d b 11, a 12 \u003d b 12, a 21 \u003d b 21, a 22 \u003d b 22

Matrices of a special kind

square matrix called upper triangular, if at i>j, and lower triangular, if at i

General view of triangular matrices:

Note that among the diagonal elements may have zero elements. Matrix is called an upper trapezoid if the following three conditions are met:

1. for i>j;

2. There is a natural number r satisfying the inequalities , what .

3. If any diagonal element , then all elements of the i-th row and all subsequent rows are equal to zero.

General view of the upper trapezoidal matrices:

at .

at .

for r=n

at r=m=n.

Note that for r=m=n, the upper trapezoidal matrix is ​​a triangular matrix with non-zero diagonal entries.

27) Actions with matrices

Matrix addition

Matrices of the same size can be added.

The sum of two such matrices A and B is the matrix C, the elements of which are equal to the sum of the corresponding elements of the matrices A and B. Symbolically, we will write as follows: A+B=C.

It is easy to see that matrix addition obeys commutative and associative laws:

(A+B)+C=A+(B+C).

The zero matrix when adding matrices performs the role of the usual zero when adding numbers: A+0=A.

Matrix subtraction.

The difference of two matrices A and B of the same size is a matrix C such that

From this definition it follows that the elements of the matrix C are equal to the difference of the corresponding elements of the matrices A and B.

The difference of matrices A and B is denoted as follows: C \u003d A - B.

3. Matrix multiplication

Consider the rule for multiplying two square matrices of the second order.

The product of matrix A and matrix B is the matrix C=AB.

Rules for multiplying rectangular matrices:

Multiplying matrix A by matrix B makes sense if the number of columns in matrix A is the same as the number of rows in matrix B.

As a result of multiplying two rectangular matrices, a matrix is ​​obtained that contains as many rows as there were rows in the first matrix and as many columns as there were columns in the second matrix.

4. Multiplying a matrix by a number

When multiplying the matrix A by the number , all the numbers that make up the matrix A are multiplied by the number . For example, let's multiply the matrix by the number 2. We get, i.e. when multiplying a matrix by a number, the factor is “introduced” under the sign of the matrix.

Matrix transposition

The transposed matrix is ​​the matrix AT obtained from the original matrix A by replacing rows with columns.

Formally, the transposed matrix for an m*n matrix A is an n*m matrix AT, defined as AT = A .

For example,

Properties of transposed matrices

2. (A + B)T = AT + BT

28) The concept of the determinant of the nth order

Let there be a square table consisting of numbers arranged in n horizontal and n vertical rows. Using these numbers, according to certain rules, a certain number is calculated, which is called the determinant of the nth order and denoted as follows:

(1)

Horizontal rows in the determinant (1) are called rows, vertical rows are called columns, and numbers are called elements of the determinant (the first index means the row number, the second one is the column number, at the intersection of which there is an element; i = 1, 2, ..., n; j = 1, 2, ..., n). The order of a determinant is the number of its rows and columns.

An imaginary straight line connecting the elements of the determinant for which both indices are the same, i.e. elements

is called the main diagonal, the other diagonal is called the side diagonal.

The determinant of the nth order is a number that is the algebraic sum of n! terms, each of which is the product of n of its elements, taken only one from each n rows and from each n columns of a square table of numbers, with half of the (certain) terms taken with their signs, and the rest with opposite signs.

Let us show how the determinants of the first three orders are calculated.

The first order determinant is the element itself i.e.

The second order determinant is the number obtained as follows:

(2)

Formula (3) shows that with their signs, terms are taken that are the product of elements of the main diagonal, as well as elements located at the vertices of two triangles, the bases of which are parallel to it; with opposite - terms that are products of elements of the secondary diagonal, as well as elements located at the vertices of two triangles that are parallel to it.

Example 2. Calculate the third order determinant:

Solution. Using the rule of triangles, we get

The calculation of the determinants of the fourth and subsequent orders can be reduced to the calculation of the determinants of the second and third orders. This can be done using the properties of the determinants. We now turn to their consideration.

Properties of the nth order determinant

Property 1. When replacing rows with columns (transposing), the value of the determinant will not change, i.e.

Property 2. If at least one row (row or column) consists of zeros, then the determinant is equal to zero. The proof is obvious.

Indeed, then in each term of the determinant one of the factors will be zero.

Property 3. If two adjacent parallel rows (rows or columns) are interchanged in the determinant, then the determinant will change sign to the opposite, i.e.

Property 4. If there are two identical parallel rows in the determinant, then the determinant is equal to zero:

Property 5. If two parallel rows in the determinant are proportional, then the determinant is equal to zero:

Property 6. If all elements of the determinant in the same row are multiplied by the same number, then the value of the determinant will change this number of times:

Consequence. The common factor contained in all elements of the same row can be taken out of the determinant sign, for example:

Property 7. If in the determinant all elements of one row are presented as the sum of two terms, then it is equal to the sum of two determinants:

Property 8. If we add the product of the corresponding elements of a parallel series by a constant factor to the elements of any series, then the value of the determinant will not change:

Property 9. If we add a linear combination of the corresponding elements of several parallel rows to the elements of the i-th row, then the value of the determinant will not change:

it is possible to construct various minors of the first, second and third order.