Lines of the second order.
Ellipse and its canonical equation. Circle
After thorough study straight lines in the plane We continue to study the geometry of the two-dimensional world. The stakes are doubled and I invite you to visit a picturesque gallery of ellipses, hyperbolas, parabolas, which are typical representatives second order lines. The excursion 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 ( – real number, – 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's and Y's in non-negative integers degrees.
Line order 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 ease of existence, we assume that all subsequent calculations take place in Cartesian coordinates.
General equation the second order line has the form , where 
– arbitrary real numbers (It is customary to write it with a factor of two), and the coefficients are not equal to zero at the same time.
If , then the equation simplifies to 
, and if the coefficients are not equal to zero at the same time, then this is exactly general equation of a “flat” line, which represents first order line.
Many have understood the meaning of the new terms, but, nevertheless, in order to 100% master the material, we stick our fingers into the socket. To determine the line order, you need to iterate all terms its equations and find for each of them sum of degrees incoming variables.
For example:
the term contains “x” to the 1st power;
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 defines the line second order:
the term contains “x” to the 2nd power;
the summand has the sum of the powers of the variables: 1 + 1 = 2;
the term contains “Y” to the 2nd power;
all other terms - less degrees.
Maximum value: 2
If we additionally add, say, to our equation, then it will already determine third-order line. It is obvious that the general form of the 3rd order line equation contains a “full set” of terms, the sum of the powers of the variables in which is equal to three:
, where the coefficients are not equal to zero at the same time.
In the event that you add one or more suitable terms that contain 
, then we will already talk about 4th order lines, etc.
We will have to encounter algebraic lines of the 3rd, 4th and higher orders more than once, in particular, when getting acquainted with polar coordinate system.
However, let's return to the general equation and remember its simplest school variations. As examples, a parabola arises, the equation of which can be easily reduced to a general form, and a hyperbola with an equivalent equation. However, not everything is so smooth...
Significant disadvantage general equation is that it is almost always not clear which line it sets. Even in the simplest case, you won’t immediately realize that this is a hyperbole. Such layouts are good only at a masquerade, so a typical problem is considered in the course of analytical geometry bringing the 2nd order line equation to canonical form.
What is the canonical form of an equation?
This is generally accepted standard view 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 easily visible.
It is obvious that any 1st order line is a straight line. On the second floor, it is no longer the watchman who is waiting for us, but a much more diverse company of nine statues:
Classification of second order lines
Using a special set of actions, any equation of a second-order line is reduced to one of the following forms:
( and are positive real numbers)
1) 
– canonical equation of the ellipse;
2) – canonical equation of a hyperbola;
3) 
– canonical equation of a parabola;
4) – imaginary ellipse;
5) – a pair of intersecting lines;
6) – pair imaginary intersecting lines (with a single valid point of intersection at the origin);
7) – a pair of parallel lines;
8) – pair imaginary parallel lines;
9) – a pair of coincident lines.
Some readers may have the impression that the list is incomplete. For example, in point No. 7, the equation specifies the pair direct, parallel to the axis, and the question arises: where is the equation that determines the lines parallel to the ordinate axis? Answer: it not considered canonical. Straight lines represent the same standard case, rotated by 90 degrees, and the additional entry in the classification is redundant, since it does not bring anything fundamentally new.
Thus there are nine and only nine various types lines of the 2nd order, but in practice they are most often found ellipse, hyperbola and parabola.
Let's look at the ellipse first. As usual, I focus on those points that have great importance to solve 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 “the eccentricity of an ellipse”.
The canonical equation of an ellipse has the form , where are positive real numbers, and . I will formulate the very definition of an ellipse later, but for now it’s time to take a break from the talking shop and solve a common problem:
How to build an ellipse?
Yes, just take it and just draw it. The task occurs frequently, and a significant part of students do not cope with the drawing correctly:
Example 1
Construct the ellipse given by the equation
Solution: First, let’s bring the equation to canonical form: 
Why bring? One of the advantages of the canonical equation is that it allows you to instantly determine vertices of the ellipse, which are located at 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 shaft ellipse;
number 
– minor axis.
in our example: .
To quickly imagine what a particular ellipse looks like, just look at the values of “a” and “be” of its canonical equation.
Everything is fine, smooth and beautiful, but there is one caveat: I made the drawing using the program. And you can make the drawing using any application. However, in harsh reality, there is a checkered piece of paper on the table, and mice dance in circles on our hands. People with artistic talent, of course, can argue, but you also have mice (though smaller ones). It’s not in vain that humanity invented the ruler, compass, 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. It’s all right if the ellipse is small, for example, with semi-axes. Alternatively, you can reduce the scale and, accordingly, the dimensions of the drawing. But in general, it is highly desirable to find additional points.
There are two approaches to constructing an ellipse - geometric and algebraic. I don’t like construction using a compass and ruler because the algorithm is not the shortest and the drawing is significantly cluttered. 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 equation of the ellipse in the draft we quickly express: 
The equation then breaks down into two functions: 
– defines the upper arc of the ellipse; 
– defines the bottom arc of the ellipse.
The ellipse defined by the canonical equation is symmetrical with respect to the coordinate axes, as well as with respect to the origin. And this is great - symmetry is almost always a harbinger of freebies. Obviously, it is enough to deal with the 1st coordinate quarter, so we need the function 
. It begs to be found for additional points with abscissas 
. Let's tap three SMS messages on the calculator: 
Of course, it’s also nice that if a serious mistake is made in the calculations, it will immediately become clear during construction.
Let's mark the points on the drawing (red), symmetrical points on the remaining arcs (blue) and carefully connect the entire company with a line: 
It is better to draw the initial sketch very thinly, and only then apply pressure with a pencil. The result should be a quite decent ellipse. By the way, would you like to know what this curve is?
Definition of an ellipse. Ellipse foci and ellipse eccentricity
Ellipse is special case oval The word “oval” should not be understood in the philistine sense (“the child drew an oval”, etc.). This is a mathematical term that has a detailed formulation. The purpose of this lesson is not to consider the theory of ovals and their various types, which receive virtually no attention in the standard course of analytical geometry. And, in accordance with more current needs, we immediately move on to the strict definition of an ellipse:
Ellipse 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 quantity, numerically equal to length major axis of this ellipse: .
In this case, the distances between the focuses are less than this value: .
Now everything will become clearer: 
Imagine that blue dot“travels” along 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 “um” at the right vertex of the ellipse, then: , which is what needed to be checked.
Another way of drawing it is based on the definition of an ellipse. Higher mathematics is sometimes the cause of tension and stress, so it’s time to have another unloading session. Please take whatman 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 pencil lead will end up at a certain point that belongs to the ellipse. Now start drawing the pencil along the sheet of paper, keeping the green thread tightly taut. Continue the process until you return to the starting point... great... the drawing can be checked by the doctor and teacher =)
How to find the foci of an ellipse?
In the above example, I depicted “ready-made” focal points, and now we will learn how to extract them from the depths of geometry.
If an ellipse is given by a canonical equation, then its foci have coordinates 
, where is it distance from each focus to the center of symmetry of the ellipse.
The calculations are simpler than simple: 
! The specific coordinates of foci cannot be identified with the meaning of “tse”! I repeat that 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 also cannot be tied to the canonical position of the ellipse. 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 consider this moment during further study of the topic.
Ellipse eccentricity and its geometric meaning
The eccentricity of an ellipse is a ratio that can take values within the range .
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 semimajor axis will remain constant. Then the eccentricity formula will take the form: .
Let's start bringing the eccentricity value closer to unity. This is only possible if . What does it mean? ...remember the tricks 
. This means that the foci of the ellipse will “move apart” 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.
Thus, the closer the ellipse eccentricity value is to unity, the more elongated the ellipse.
Now let's model the opposite process: the foci of the ellipse 
walked towards each other, approaching the center. This means that the value of “ce” becomes less and less and, accordingly, the eccentricity tends to zero: .
In this case, the “green segments” will, on the contrary, “become crowded” and they will begin to “push” the ellipse line up and down.
Thus, The closer the eccentricity value is to zero, the more similar the ellipse is to... 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 semi-axes, the canonical equation of the ellipse takes the form , which reflexively transforms to the equation of a circle with a center at the origin of radius “a”, well known from school.
In practice, the notation with the “speaking” letter “er” is more often used: . The radius is the length of a segment, with each point of the circle removed from the center by a radius distance.
Note that the definition of an ellipse remains completely correct: the foci coincide, and the sum of the lengths of the coincident segments for each point on the circle is a constant. Since the distance between the foci is , then the eccentricity of any circle is zero.
Constructing a circle is easy and quick, just use 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 the cheerful Matanov form:
– function of the upper semicircle;
– function of the lower semicircle.
Then we find the required values, differentiate, integrate and do other good things.
The article, of course, is for reference only, but how can you live in the world without love? Creative task For independent decision
Example 2
Compose the canonical equation of an ellipse if one of its foci and semi-minor axis are known (the center is at the origin). Find vertices, additional points and draw a line in the drawing. Calculate eccentricity.
Solution and drawing at the end of the lesson
Let's add an action:
Rotate and parallel translate an ellipse
Let's return to the canonical equation of the ellipse, namely, to the condition, the mystery of which has tormented inquisitive minds since the first mention of this curve. So we looked at the ellipse 
, but isn’t it possible in practice to meet the equation 
? After all, here, however, it seems to be an ellipse too!
This kind of equation is rare, but it does come across. And it actually defines an ellipse. Let's demystify: 
As a result of the construction, our native ellipse was obtained, rotated by 90 degrees. That is, 
- This non-canonical entry ellipse 
. Record!- the equation 
does not define any other ellipse, since there are no points (foci) on the axis that would satisfy the definition of an ellipse.
The canonical equation of the ellipse has the form
where a is the semimajor axis; b – semi-minor axis. The points F1(c,0) and F2(-c,0) − c are called
a, b - semi-axes of the ellipse.
Finding the foci, eccentricity, directrixes of an ellipse, if its canonical equation is known.
Definition of hyperbole. Hyperbole tricks.
Definition. A hyperbola is a set of points on 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 – focuses of the hyperbola. F1F2 = 2c.
The canonical equation of a hyperbola. Semi-axes of a hyperbola. Constructing a hyperbola if its canonical equation is known.
Canonical equation:
The semimajor axis of a hyperbola is half the minimum distance between two branches of the hyperbola, on the positive and negative sides axes (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 through the conic section and eccentricity, then the expression will take the form:
Finding the foci, eccentricity, directrixes of a hyperbola, if its canonical equation is known.
Hyperbola eccentricity
Definition. The ratio is called the eccentricity of the hyperbola, where c –
half the distance between the foci, and is the real semi-axis.
Taking into account the fact that c2 – a2 = b2:
If a = b, e = , then the hyperbola is called equilateral (equilateral).
Directrixes of a hyperbole
Definition. Two straight lines perpendicular to the real axis of the hyperbola and located symmetrically relative to the center at a distance a/e from it are called directrixes of the hyperbola. Their equations are: .
Theorem. If r is the distance from an arbitrary point M of the hyperbola to any 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 about which we're talking about in the definition, is called the focus of the parabola, and the straight line is its directrix.
Canonical equation of a parabola. Parabola parameter. Construction of a parabola.
The canonical equation of a parabola in rectangular system coordinates: (or if the axes are swapped).
Construction of a parabola at given value parameter p is executed in the following sequence:
Draw the axis of symmetry of the parabola and plot the segment KF=p on it;
Directrix DD1 is drawn through point K perpendicular to the axis of symmetry;
The segment KF is divided in half to obtain vertex 0 of the parabola;
A series of arbitrary points 1, 2, 3, 5, 6 are measured from the top with a gradually increasing distance between them;
Through these points, draw auxiliary straight lines perpendicular to the axis of the parabola;
On auxiliary 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.
Theorem. In the canonical coordinate system for an ellipse, the equation of the ellipse has the form:
Proof. We 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 the 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 this and the definition of the equation of a curve it will follow that equation (4) is an 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:
Let's use the formula for the distance between two points on the coordinate plane and use this formula to find the focal radii of a given point M:
Where do we get it from:
Let's move one root to the right side of the equality and square it:
Reducing, we get:
We present similar ones, reduce by 4 and remove the radical:
.
Squaring
Open the brackets and shorten by:
where we get:
Using equality (2), we obtain:
.
Dividing the last equality by , we obtain equality (4), etc.
2) Let now a pair of numbers (x, y) satisfy equation (4) and let M(x, y) be the corresponding point on the coordinate plane Oxy.
Then from (4) it follows:
We substitute this equality into the expression for the focal radii of point M:
.
Here we used equality (2) and (3).
Thus, . Likewise, .
Now note that from equality (4) it follows that
Or etc. , then the inequality follows:
From here it follows, in turn, that
From equalities (5) it follows 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 an 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 is called the geometric locus of points in a 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 interested can become acquainted with them in more serious textbooks on analytical geometry. Here we only note that an ellipse is a curve obtained as a projection onto the plane of a circle lying in the plane that forms sharp corner with a plane. Unlike a circle, it is not possible to write down the equation of an ellipse in an arbitrary coordinate system in a “convenient” form. Therefore, for a fixed ellipse, it is necessary to select a coordinate system so that its equation is quite simple. Let and be the foci of the ellipse. Let's place the origin of the coordinate system at the middle of the segment. The axis is directed along this segment, the axis is directed perpendicular to this segment
24)Hyperbola
From a school mathematics course we know that a curve defined by the equation , where is a number, is called a hyperbola. However, this is a special case of a hyperbola (equilateral hyperbola). Definition 12. 5 A hyperbola is the locus of points on a plane, for each of which absolute value the difference in distances to two fixed points of the same plane, called the foci of a hyperbola, is a constant value. Just as in the case of an ellipse, to obtain the equation of a hyperbola, we choose a suitable coordinate system. Let's place the origin of coordinates in the middle of the segment between the foci, direct the axis along this segment, and direct the ordinate axis perpendicular to it. Theorem 12. 3 Let the distance between the foci and the hyperbola be equal, and the absolute value of the difference in the distances from the point of the hyperbola to the foci be equal. 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 focuses are , . Using formula (10.4) for the case of a plane, we obtain By definition of a hyperbola We write this equation in the form We square both sides: After bringing similar terms and dividing by 4, we arrive at the equality 
Again, we square both sides: Opening the bracket and bringing similar terms, we get Taking into account formula (12.9), the equation takes the form 
Let's divide both sides of the equation by and get equation (12.8) Equation (12.8) is called the canonical equation of a hyperbola. Proposition 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 axes of symmetry are
coordinate axes and , and the origin is the center of symmetry of the hyperbola. Proof. The proof is similar to Proposition 12.1. Let us construct the hyperbola given by equation (12.8). Note that, due to symmetry, it is sufficient to construct the curve only at the first coordinate angle. Let us express from the canonical equation as a function, provided that,
and build a graph of 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 (without corners). Second derivative 
is negative at all points of the interval, therefore, the graph is convex upward. Let's check the graph for the presence of an asymptote at . Let the asymptote have the equation . Then according to the rules mathematical analysis 
We multiply the expression under the limit sign and divide by . We get: So, the graph of the function has an asymptote. From the symmetry of the hyperbola it follows that it is also an asymptote. The nature of the curve in the vicinity of the point remains unclear, namely, whether the graph forms 
and the part of the hyperbola that is symmetrical to it relative to the axis at this point is an angle or a hyperbola at this point - a smooth curve (there is a tangent). To solve this issue, we express from equation (12.8) through: 
It's obvious that this function has a derivative at the point , , and at the point the hyperbola has a vertical tangent. Using the data obtained, we draw a graph of the function (Fig. 12.10). 
Rice. 12 . 10. Graph of a function Finally, using the symmetry of the hyperbola, we obtain the curve of Figure 12.11. 
Rice. 12 . 11.Hyperbole Definition 12. 6 The points of intersection of the hyperbola defined 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 ordinate axis between the points is called the imaginary axis. The numbers and are called the real and imaginary semi-axes of the hyperbola, respectively. The origin of coordinates is called its center. The quantity is called the eccentricity of the hyperbola. Note 12. 3 From equality (12.9) it follows that , that is, for the hyperbola . Eccentricity characterizes the angle between asymptotes; the closer to 1, the smaller this angle. Note 12. 4 Unlike an ellipse, in the canonical equation of a hyperbola, the relationship between the quantities and can be arbitrary. In particular, when we get an equilateral hyperbola, known from the school mathematics course. Its equation has a familiar form if we take , and the axes and direct them 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 a 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. Let's divide both sides of the equation by 4. We get the canonical equation , . We draw asymptotes and construct a hyperbola (Fig. 12.13). 
Rice. 12 . 13.Hyperbola From formula (12.9) we obtain. Then the tricks are , , . Example 12. 5 Construct a hyperbola. Find its foci and eccentricity. Solution. Let's transform the equation to the form This equation is not a canonical equation of a hyperbola, since the signs are before and opposite the signs in the canonical equation. However, if we redesignate the variables , , then in the new variables we obtain 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 an equation, that is, the equation in the original coordinates. The real semi-axis is equal to 5, the imaginary one is 2. In accordance with these data, we carry out the construction (Fig. 12.14). Rice. 12 . 14.Hyperbola with equation From formula (12.9) we obtain, , the foci lie on the real axis - , , where the coordinates are indicated in the original coordinate system.
Parabola
In the school mathematics course, the parabola was studied in sufficient detail, which, by definition, was a graph quadratic trinomial. Here we will give another (geometric) definition of a parabola. Definition 12. 7 A parabola is the geometric locus of points on 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, lower the perpendicular from the focus to the directrix. Let's place the origin of coordinates in the middle of the segment, and direct the axis along the segment so that its direction coincides with the direction of the vector. Let's draw the axis perpendicular to the axis (Fig. 12.15). 
Rice. 12 . 15 . Theorem 12. 4 Let the distance between the focus and the directrix of the parabola be equal to . 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 the equation (Fig. 12.15). Let be the current point of the parabola. Then, using formula (10.4) for the plane case, we find 
The distance from a point to the directrix is the length of the perpendicular dropped to the directrix from the point. From Figure 12.15 it is obvious that . Then by the definition of a parabola, that is 
Let's square both sides of the last equation: 
where 
After bringing similar terms, we obtain equation (12.10). Equation (12.10) is called the canonical equation of a parabola. Proposition 12. 4 A parabola has an axis of symmetry. If a parabola is given by a canonical equation, then the axis of symmetry coincides with the axis. Proof. Proceed in the same way as the proof (Propositions 12.1). The point of intersection of the axis of symmetry with the parabola is called the vertex of the parabola. If we redesignate the variables , then equation (12.10) can be written in a form that coincides with the usual parabola equation in a school mathematics course. Therefore, we will draw a parabola without additional research(Fig. 12.16). 
Rice. 12 . 16. Parabola Example 12. 6 Construct a parabola. Find her focus and director. 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 will find several points of the parabola. To do this, we assign values to the variable and find the values. Let's take points , , . Taking into account symmetry about the axis, we draw a curve (Fig. 12.17) 
Rice. 12 . 17. A parabola given by the equation Focus lies on the axis at a distance from the vertex, that is, it has coordinates . The directrix has an equation, that is, . A parabola, like an ellipse, has a property associated with the reflection of light (Fig. 12.18). Let us formulate the property again without proof. Proposition 12. 5 Let be the focus of the parabola, an arbitrary point of the parabola, and a ray with its origin at a point parallel to the axis of the parabola. Then the normal to the parabola at the point divides the angle formed by the segment and the ray in half. 
Rice. 12 . 18. Reflection of a light ray from a parabola This property means that a ray of light leaving the focus, reflected from the parabola, will then go parallel to the axis of this parabola. And vice versa, all rays coming from infinity and parallel to the axis of the parabola will converge at its focus. This property is widely used in technology. Spotlights usually have a mirror, the surface of which is obtained by rotating a parabola around its axis of symmetry (parabolic mirror). The light source in spotlights is placed at the focus of a parabola. As a result, the spotlight produces a beam of almost parallel rays of light. The same property is used in receiving antennas for space communications and in telescope mirrors, which collect a stream of parallel rays of radio waves or a stream of parallel rays 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 matrix concepts: 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 the designation appears:
However, to briefly denote a matrix, one large 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 included in this matrix are called its elements. In the entry a ij, the first index i means the row number, and the second index j means the column number.
For example, matrix
this is a matrix of order 2×3, its elements are a 11 =1, a 12 =x, a 13 =3, a 21 =-2y, ...
So, we have introduced the definition of a matrix. Let us consider the types of matrices and give the corresponding definitions.
Types of matrices
Let us introduce the concept of matrices: square, diagonal, unit and zero.
Definition of a square matrix: Square matrix An n-th order matrix is called an n×n matrix.
In the case of a square matrix
The concept of main and secondary diagonals is introduced. Main diagonal of a matrix is the diagonal that goes from the upper left corner of the matrix to its 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: Single(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 whose elements are all zero.
Two matrices A and B are said to be equal (A=B) if they are the same size (that is, they have the same number of rows and the same number of columns and their corresponding elements are equal). So, if 
then A=B, if a 11 =b 11, a 12 =b 12, a 21 =b 21, a 22 =b 22
Matrices of a special type
Square matrix 
called upper triangular, if at i>j, And lower triangular, if at i
General form triangular matrices:
Note that among the diagonal elements 
there may be elements equal to zero. Matrix 
called upper trapezius if the following three conditions are met:
1. for i>j;
2. There is such a thing natural number r satisfying the inequalities 
, What 
.
3. If any diagonal element is , then all i-th elements line and all subsequent lines are zero.
General view of the upper trapezoidal matrices:
at 
.
at .
at r=n
at r=m=n.
Note that when r=m=n, the upper trapezoidal matrix is a triangular matrix with non-zero diagonal elements.
27) Actions with matrices
Matrix addition
Matrices of the same size can be stacked.
The sum of two such matrices A and B is called matrix C, the elements of which are equal to the sum of the corresponding elements of matrices A and B. Symbolically, we will write it like this: A+B=C.
It is easy to see that the addition of matrices obeys the commutative and combinational laws:
(A+B)+C=A+(B+C).
When adding matrices, the zero matrix plays the role of an ordinary zero when adding numbers: A+0=A.
Subtraction of matrices.
The difference between two matrices A and B of the same size is a matrix C such that
From this definition it follows that the elements of matrix C are equal to the difference of the corresponding elements of matrices A and B.
The difference between matrices A and B is denoted as follows: C=A – B.
3. Matrix multiplication
Consider the rule for multiplying two second-order square matrices.
The product of matrix A and matrix B is called matrix C=AB.
Rules for multiplying rectangular matrices:
Multiplying matrix A by matrix B makes sense in the case when the number of columns of matrix A coincides with the number of rows in matrix B.
As a result of multiplying two rectangular matrices, a matrix is obtained containing 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 matrix A is multiplied by number , all numbers that make up matrix A are multiplied by 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 Transpose
Transposed matrix is a matrix AT obtained from the original matrix A by replacing rows with columns.
Formally, the transposed matrix for a matrix A of dimensions m*n is a matrix AT of dimensions n*m, defined as AT = A .
For example,
Properties of transposed matrices
2. (A + B)T = AT + BT
28) The concept of nth order determinant
Let us be given 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 nth order determinant and is denoted as follows:
(1)
The horizontal rows in the determinant (1) are called rows, the vertical rows are called columns, the numbers are elements of the determinant (the first index means the row number, the second – the column number at the intersection of which the element stands; 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 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 secondary diagonal.
A determinant of 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 the terms taken with their signs are the product of the elements of the main diagonal, as well as the elements located at the vertices of two triangles, the bases of which are parallel to it; with opposite ones - terms that are products of elements of the side 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 triangle rule, we get
The calculation of determinants of the fourth and subsequent orders can be reduced to the calculation of determinants of the second and third orders. This can be done using the properties of determinants. We now move on to consider them.
Properties of the nth order determinant
Property 1. When replacing rows with columns (transposition), 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.
In fact, 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 swapped in the determinant, then the determinant will change its sign to the opposite, i.e.
Property 4. If the determinant contains two identical parallel series, then the determinant is equal to zero:
Property 5. If two parallel series in the determinant are proportional, then the determinant is equal to zero:
Property 6. If all elements of the determinant that are in the same row are multiplied by the same number, then the value of the determinant will change by this number of times:
Consequence. The common factor contained in all elements of one row can be taken out of the determinant sign, for example:
Property 7. If in a determinant all elements of one series are presented as the sum of two terms, then it equal to the sum two qualifiers:
Property 8. If the product of the corresponding elements of a parallel series by a constant factor is added to the elements of any series, then the value of the determinant will not change:
Property 9. If a linear combination of the corresponding elements of several parallel series is added to the elements of the i-th series, then the value of the determinant will not change:
one can construct various minors of the first, second and third order.
Second order curves on a plane are lines defined by equations in which the variable coordinates x And y are contained in the second degree. These include the ellipse, hyperbola and parabola.
The general form of the second order curve equation is as follows:
Where A, B, C, D, E, F- numbers and at least one of the coefficients A, B, C not equal to zero.
When solving problems with second-order curves, the canonical equations of the ellipse, hyperbola and parabola are most often considered. It is easy to move on to them from general equations; example 1 of problems with ellipses will be devoted to this.
Ellipse given by the canonical equation
Definition of an ellipse. An ellipse is the set of all points of the plane for which the sum of the distances to the points called foci is a constant value greater than the distance between the foci.
The focuses are indicated as in the figure below.
The canonical equation of an ellipse has the form:
Where a And b (a > b) - the lengths of the semi-axes, i.e., half the lengths of the segments cut off by the ellipse on the coordinate axes.
The straight line passing through the foci of the ellipse is its axis of symmetry. Another axis of symmetry of an ellipse is a straight line passing through the middle of a segment perpendicular to this segment. Dot ABOUT the intersection of these lines serves as the center of symmetry of the ellipse or simply the center of the ellipse.
The abscissa axis of the ellipse intersects at the points ( a, ABOUT) And (- a, ABOUT), and the ordinate axis is in points ( b, ABOUT) And (- b, ABOUT). These four points are called the vertices of the ellipse. The segment between the vertices of the ellipse on the x-axis is called its major axis, and on the ordinate axis - its minor axis. Their segments from the top to the center of the ellipse are called semi-axes.
If a = b, then the equation of the ellipse takes the form . This is the equation of a circle with radius a, and a circle is a special case of an ellipse. An ellipse can be obtained from a circle of radius a, if you compress it into a/b times along the axis Oy .
Example 1. Check if a line given by a general equation is 
, ellipse.
Solution. We transform the general equation. We use the transfer of the free term to the right side, the term-by-term division of the equation by the same number and the reduction of fractions:
Answer. The equation obtained as a result of the transformations is the canonical equation of the ellipse. Therefore, this line is an ellipse.
Example 2. Compose the canonical equation of an ellipse if its semi-axes are 5 and 4, respectively.
Solution. We look at the formula for the canonical equation of an ellipse and substitute: the semimajor axis is a= 5, the semiminor axis is b= 4 . We obtain the canonical equation of the ellipse:
Points and , indicated in green on the major axis, where
are called tricks.
called eccentricity ellipse.
Attitude b/a characterizes the “oblateness” of the ellipse. The smaller this ratio, the more the ellipse is elongated along the major axis. However, the degree of elongation of an ellipse is more often expressed through eccentricity, the formula for which is given above. For different ellipses, the eccentricity varies from 0 to 1, always remaining less than unity.
Example 3. Compose the canonical equation of the ellipse if the distance between the foci is 8 and the major axis is 10.
Solution. Let's make some simple conclusions:
If the major axis is equal to 10, then its half, i.e. the semi-axis a = 5 ,
If the distance between the foci is 8, then the number c of the focal coordinates is equal to 4.
We substitute and calculate:
The result is the canonical equation of the ellipse:
Example 4. Compose the canonical equation of an ellipse if its major axis is 26 and its eccentricity is .
Solution. As follows from both the size of the major axis and the eccentricity equation, the semimajor axis of the ellipse a= 13. From the eccentricity equation we express the number c, needed to calculate the length of the minor semi-axis:
.
We calculate the square of the length of the minor semi-axis:
We compose the canonical equation of the ellipse:
Example 5. Determine the foci of the ellipse given by the canonical equation.
Solution. Find the number c, which determines the first coordinates of the ellipse's foci:
.
We get the focuses of the ellipse:
Example 6. The foci of the ellipse are located on the axis Ox symmetrically about the origin. Compose the canonical equation of the ellipse if:
1) the distance between the focuses is 30, and the major axis is 34
2) minor axis 24, and one of the focuses is at point (-5; 0)
3) eccentricity, and one of the foci is at point (6; 0)
Let's continue to solve ellipse problems together
If is an arbitrary point of the ellipse (indicated in green in the upper right part of the ellipse in the drawing) and is the distance to this point from the foci, then the formulas for the distances are as follows:
For each point belonging to the ellipse, the sum of the distances from the foci is a constant value equal to 2 a.
Lines defined by equations
are called headmistresses ellipse (in the drawing there are red lines along the edges).
From the two equations above it follows that for any point of the ellipse
,
where and are the distances of this point to the directrixes and .
Example 7. Given an ellipse. Write an equation for its directrixes.
Solution. We look at the directrix equation and find that we need to find the eccentricity of the ellipse, i.e. We have all the data for this. We calculate:
.
We obtain the equation of the directrixes of the ellipse:
Example 8. Compose the canonical equation of an ellipse if its foci are points and directrixes are lines.
