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 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% assimilate 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 an example, a parabola suggests itself, the equation of which can easily be reduced to general appearance, and a hyperbola with the 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 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 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 an 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 the question of finding 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 us mark points in the drawing (red), symmetrical points on the remaining arcs ( Blue colour) and carefully connect the whole 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

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 that has 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 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 the length of the major axis of this ellipse: .
At the same time, the distances between focuses are smaller given value: .

Now everything will become clearer:

Imagine that the 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, sometimes the cause of tension and stress, so it’s time to conduct another unloading session. Please take whatman paper or large leaf cardboard and nail 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 moving the pencil along the piece of paper, keeping the green thread 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.

Eccentricity of the ellipse 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 torments 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.

11.1. Basic Concepts

Let's consider lines defined by equations of the second degree relative to the current coordinates

The coefficients of the equation are real numbers, but at least one of the numbers A, B, or C is nonzero. Such lines are called lines (curves) of the second order. Below it will be established that equation (11.1) defines a circle, ellipse, hyperbola or parabola on the plane. Before moving on to this statement, let us study the properties of the listed curves.

11.2. Circle

The simplest second-order curve is a circle. Recall that a circle of radius R with center at a point is the set of all points M of the plane satisfying the condition . Let a point in a rectangular coordinate system have coordinates x 0, y 0 and - an arbitrary point on the circle (see Fig. 48).

Then from the condition we obtain the equation

(11.2)

Equation (11.2) is satisfied by the coordinates of any point on a given circle and is not satisfied by the coordinates of any point not lying on the circle.

Equation (11.2) is called canonical equation of a circle

In particular, setting and , we obtain the equation of a circle with center at the origin .

The circle equation (11.2) after simple transformations will take the form . When comparing this equation with the general equation (11.1) of a second-order curve, it is easy to notice that two conditions are satisfied for the equation of a circle:

1) the coefficients for x 2 and y 2 are equal to each other;

2) there is no member containing the product xy of the current coordinates.

Let's consider the inverse problem. Putting the values ​​and in equation (11.1), we obtain

Let's transform this equation:

(11.4)

It follows that equation (11.3) defines a circle under the condition . Its center is at the point

.

, and the radius If

.

, then equation (11.3) has the form It is satisfied by the coordinates of a single point

. , then equation (11.4), and therefore the equivalent equation (11.3), will not define any line, since the right side of equation (11.4) is negative, and the left is not negative (say: “an imaginary circle”).

11.3. Ellipse

Canonical ellipse equation

Ellipse is the set of all points of a plane, the sum of the distances from each of which to two given points of this plane, called tricks , is a constant value greater than the distance between the foci.

Let us denote the focuses by F 1 And F 2, the distance between them is 2 c, and the sum of distances from an arbitrary point of the ellipse to the foci - in 2 a(see Fig. 49). By definition 2 a > 2c, i.e. a > c.

To derive the equation of the ellipse, we choose a coordinate system so that the foci F 1 And F 2 lay on the axis, and the origin coincided with the middle of the segment F 1 F 2.

Then the foci will have the following coordinates: and .

Let be an arbitrary point of the ellipse. Then, according to the definition of an ellipse, i.e.

This, in essence, is the equation of an ellipse. Let us transform equation (11.5) to more simple view

in the following way: a>Because With

(11.6)

, That . Let's put

(11.7)

Then the last equation will take the form or It can be proven that equation (11.7) is equivalent to the original equation. It's called .

canonical ellipse equation

An ellipse is a second order curve.

Studying the shape of an ellipse using its equation

Let us establish the shape of the ellipse using its canonical equation.

1. Equation (11.7) contains x and y only in even powers, so if a point belongs to an ellipse, then the points ,, also belong to it. It follows that the ellipse is symmetrical with respect to the and axes, as well as with respect to the point, which is called the center of the ellipse. 1 , 2. Find the points of intersection of the ellipse with the coordinate axes. Putting , we find two points and , at which the axis intersects the ellipse (see Fig. 50). Putting in equation (11.7) , we find the points of intersection of the ellipse with the axis: and . Points , A, A 2 B 1 B 2 are called It follows that the ellipse is symmetrical with respect to the and axes, as well as with respect to the point, which is called the center of the ellipse. 1 2. Find the points of intersection of the ellipse with the coordinate axes. Putting , we find two points and , at which the axis intersects the ellipse (see Fig. 50). Putting in equation (11.7) , we find the points of intersection of the ellipse with the axis: and . Points And vertices of the ellipse. Segments a B 1 B 2 , as well as their lengths 2 and 2 b are called accordingly a And , as well as their lengths 2 major and minor axes ellipse. Numbers are called large and small respectively

axle shafts

4. In equation (11.7), the sum of non-negative terms and is equal to one. Consequently, as one term increases, the other will decrease, i.e. if it increases, it decreases and vice versa.

From the above it follows that the ellipse has the shape shown in Fig. 50 (oval closed curve).

More information about the ellipse

The shape of the ellipse depends on the ratio.

When the ellipse turns into a circle, the equation of the ellipse (11.7) takes the form . The ratio is often used to characterize the shape of an ellipse.<ε< 1, так как 0<с<а. С учетом равенства (11.6) формулу (11.8) можно переписать в виде

The ratio of half the distance between the foci to the semi-major axis of the ellipse is called the eccentricity of the ellipse and o6o is denoted by the letter ε (“epsilon”):

with 0

This shows that the smaller the eccentricity of the ellipse, the less flattened the ellipse will be; if we set ε = 0, then the ellipse turns into a circle.

Let M(x;y) be an arbitrary point of the ellipse with foci F 1 and F 2 (see Fig. 51). The lengths of the segments F 1 M = r 1 and F 2 M = r 2 are called the focal radii of the point M. Obviously,

The formulas hold Direct lines are called

Theorem 11.1. .

If is the distance from an arbitrary point of the ellipse to some focus, d is the distance from the same point to the directrix corresponding to this focus, then the ratio is a constant value equal to the eccentricity of the ellipse:

From equality (11.6) it follows that . If, then equation (11.7) defines an ellipse, the major axis of which lies on the Oy axis, and the minor axis on the Ox axis (see Fig. 52). The foci of such an ellipse are at points and , where

11.4. Hyperbola Canonical hyperbola equation tricks Hyperbole

Let us denote the focuses by F 1 And F 2 is the set of all points of the plane, the modulus of the difference in distances from each of them to two given points of this plane, called , is a constant value less than the distance between the foci. the distance between them is 2s, and the modulus of the difference in distances from each point of the hyperbola to the foci through 2s < , is a constant value less than the distance between the foci. 2a a < c.

. A-priory F 1 And , i.e. To derive the hyperbola equation, we choose a coordinate system so that the foci F 1 F 2 F 2

lay on the axis, and the origin coincided with the middle of the segment (see Fig. 53). Then the foci will have coordinates and Let be an arbitrary point of the hyperbola. Then, according to the definition of a hyperbola

(11.9)

(11.10)

or , i.e. After simplifications, as was done when deriving the equation of the ellipse, we obtain

canonical hyperbola equation

A hyperbola is a line of the second order.

1. Equation (11.9) contains x and y only in even powers. Consequently, the hyperbola is symmetrical about the axes and , as well as about the point, which is called

the center of the hyperbola.

2. Find the points of intersection of the hyperbola with the coordinate axes. Putting in equation (11.9), we find two points of intersection of the hyperbola with the axis: and. Putting in (11.9), we get , which cannot be. Therefore, the hyperbola does not intersect the Oy axis. The points are called peaks

hyperbolas, and the segment real axis , line segment - real semi-axis

hyperbole. The segment connecting the points is called imaginary axis , number b - imaginary semi-axis 2s And . Rectangle with sides 2b .

called

basic rectangle of hyperbola

3. From equation (11.9) it follows that the minuend is not less than one, i.e., that or .

This means that the points of the hyperbola are located to the right of the line (right branch of the hyperbola) and to the left of the line (left branch of the hyperbola).

4. From equation (11.9) of the hyperbola it is clear that when it increases, it increases. This follows from the fact that the difference maintains a constant value equal to one.

From the above it follows that the hyperbola has the form shown in Figure 54 (a curve consisting of two unlimited branches).

(11.11)

Asymptotes of a hyperbola

The straight line L is called an asymptote unbounded curve K, if the distance d from point M of curve K to this straight line tends to zero when the distance of point M along curve K from the origin is unlimited.

Figure 55 provides an illustration of the concept of an asymptote: straight line L is an asymptote for curve K. Let us show that the hyperbola has two asymptotes:

Since the straight lines (11.11) and the hyperbola (11.9) are symmetrical with respect to the coordinate axes, it is sufficient to consider only those points of the indicated lines that are located in the first quarter.

Equation of an equilateral hyperbola.

the asymptotes of which are the coordinate axes

Hyperbola (11.9) is called equilateral if its semi-axes are equal to ().

(11.12)

Its canonical equation

The asymptotes of an equilateral hyperbola have equations and, therefore, are bisectors of coordinate angles.

Let's consider the equation of this hyperbola in a new coordinate system (see Fig. 58), obtained from the old one by rotating the coordinate axes by an angle.

We use the formulas for rotating coordinate axes:

We substitute the values ​​of x and y into equation (11.12):

The equation of an equilateral hyperbola, for which the Ox and Oy axes are asymptotes, will have the form . More information about hyperbole

Eccentricity .

hyperbola (11.9) is the ratio of the distance between the foci to the value of the real axis of the hyperbola, denoted by ε:

Since for a hyperbola , the eccentricity of the hyperbola is greater than one: . Eccentricity characterizes the shape of a hyperbola. Indeed, from equality (11.10) it follows that i.e.

And From this it can be seen that the smaller the eccentricity of the hyperbola, the smaller the ratio of its semi-axes, and therefore the more elongated its main rectangle. The eccentricity of an equilateral hyperbola is . Really, From this it can be seen that the smaller the eccentricity of the hyperbola, the smaller the ratio of its semi-axes, and therefore the more elongated its main rectangle. .

Focal radii

And

for points of the right branch the hyperbolas have the form and , and for the left branch - a Direct lines are called directrixes of a hyperbola. Since for a hyperbola ε > 1, then .

This means that the right directrix is ​​located between the center and the right vertex of the hyperbola, the left - between the center and the left vertex.

The directrixes of a hyperbola have the same property as the directrixes of an ellipse.

The curve defined by the equation is also a hyperbola, the real axis 2b of which is located on the Oy axis, and the imaginary axis 2

- on the Ox axis. In Figure 59 it is shown as a dotted line.

It is obvious that hyperbolas have common asymptotes. Such hyperbolas are called conjugate.

1. In equation (11.13) the variable y appears in an even degree, which means that the parabola is symmetrical about the Ox axis; The Ox axis is the axis of symmetry of the parabola.

2. Since ρ > 0, it follows from (11.13) that . Consequently, the parabola is located to the right of the Oy axis.

3. When we have y = 0. Therefore, the parabola passes through the origin.

4. As x increases indefinitely, the module y also increases indefinitely. The parabola has the form (shape) shown in Figure 61. Point O(0; 0) is called the vertex of the parabola, the segment FM = r is called the focal radius of point M.

Equations , , ( p>0) also define parabolas, they are shown in Figure 62

It is not difficult to show that the graph of a quadratic trinomial, where , B and C are any real numbers, is a parabola in the sense of its definition given above.

11.6. General equation of second order lines

Equations of second-order curves with axes of symmetry parallel to the coordinate axes

Let us first find the equation of an ellipse with a center at the point, the axes of symmetry of which are parallel to the coordinate axes Ox and Oy and the semi-axes are respectively equal a And , as well as their lengths 2. Let us place in the center of the ellipse O 1 the beginning of a new coordinate system, whose axes and semi-axes a And , as well as their lengths 2(see Fig. 64):

Finally, the parabolas shown in Figure 65 have corresponding equations.

The equation

The equations of an ellipse, hyperbola, parabola and the equation of a circle after transformations (open brackets, move all terms of the equation to one side, bring similar terms, introduce new notations for coefficients) can be written using a single equation of the form

where coefficients A and C are not equal to zero at the same time.

The question arises: does every equation of the form (11.14) determine one of the curves (circle, ellipse, hyperbola, parabola) of the second order? The answer is given by the following theorem.

Theorem 11.2. Equation (11.14) always defines: either a circle (for A = C), or an ellipse (for A C > 0), or a hyperbola (for A C< 0), либо параболу (при А×С= 0). При этом возможны случаи вырождения: для эллипса (окружности) - в точку или мнимый эллипс (окружность), для гиперболы - в пару пересекающихся прямых, для параболы - в пару параллельных прямых.

General second order equation

Let's now consider general equation second degree with two unknowns:

It differs from equation (11.14) by the presence of a term with the product of coordinates (B¹ 0). It is possible, by rotating the coordinate axes by an angle a, to transform this equation so that the term with the product of coordinates is absent.

Using axis rotation formulas

Let's express the old coordinates in terms of the new ones:

Let us choose the angle a so that the coefficient for x" · y" becomes zero, i.e., so that the equality

Thus, when the axes are rotated by an angle a that satisfies condition (11.17), equation (11.15) is reduced to equation (11.14).

Conclusion: the general second-order equation (11.15) defines on the plane (except for cases of degeneration and decay) the following curves: circle, ellipse, hyperbola, parabola.

Note: If A = C, then equation (11.17) becomes meaningless. In this case, cos2α = 0 (see (11.16)), then 2α = 90°, i.e. α = 45°. So, when A = C, the coordinate system should be rotated by 45°.

Definition. An ellipse is the geometric locus of points on a plane, the sum of the distances of each of which from two given points of this plane, called foci, is a constant value (provided that this value is greater than the distance between the foci).

Let us denote the foci by the distance between them - by , and the constant value equal to the sum of the distances from each point of the ellipse to the foci by (by condition).

Let's construct a Cartesian coordinate system so that the foci are on the abscissa axis, and the origin of coordinates coincides with the middle of the segment (Fig. 44). Then the foci will have the following coordinates: left focus and right focus. Let us derive the equation of the ellipse in the coordinate system we have chosen. For this purpose, consider an arbitrary point of the ellipse. By definition of an ellipse, the sum of the distances from this point to the foci is equal to:

Using the formula for the distance between two points, we therefore obtain

To simplify this equation, we write it in the form

Then squaring both sides of the equation, we get

or, after obvious simplifications:

Now we square both sides of the equation again, after which we have:

or, after identical transformations:

Since, according to the condition in the definition of an ellipse, then the number is positive. Let us introduce the notation

Then the equation will take the following form:

By the definition of an ellipse, the coordinates of any of its points satisfy equation (26). But equation (29) is a consequence of equation (26). Consequently, it is also satisfied by the coordinates of any point of the ellipse.

It can be shown that the coordinates of points that do not lie on the ellipse do not satisfy equation (29). Thus, equation (29) is the equation of an ellipse. It is called the canonical equation of the ellipse.

Studying the shape of an ellipse using its equation

First of all, let's pay attention to the fact that this equation contains only even powers of x and y. This means that if any point belongs to an ellipse, then it also contains a point symmetrical with the point relative to the abscissa axis, and a point symmetrical with the point relative to the ordinate axis. Thus, the ellipse has two mutually perpendicular axes of symmetry, which in our chosen coordinate system coincide with the coordinate axes. We will henceforth call the axes of symmetry of the ellipse the axes of the ellipse, and the point of their intersection the center of the ellipse. The axis on which the foci of the ellipse are located (in this case, the abscissa axis) is called the focal axis.

Let us first determine the shape of the ellipse in the first quarter. To do this, let’s solve equation (28) for y:

It is obvious that here , since y takes imaginary values. As you increase from 0 to a, y decreases from b to 0. The part of the ellipse lying in the first quarter will be an arc bounded by points B (0; b) and lying on the coordinate axes (Fig. 45). Using now the symmetry of the ellipse, we come to the conclusion that the ellipse has the shape shown in Fig. 45.

The points of intersection of the ellipse with the axes are called the vertices of the ellipse. From the symmetry of the ellipse it follows that, in addition to the vertices, the ellipse has two more vertices (see Fig. 45).

The segments and connecting opposite vertices of the ellipse, as well as their lengths, are called the major and minor axes of the ellipse, respectively. The numbers a and b are called the major and minor semi-axes of the ellipse, respectively.

The ratio of half the distance between the foci to the semi-major axis of the ellipse is called the eccentricity of the ellipse and is usually denoted by the letter:

Since , the eccentricity of the ellipse is less than unity: Eccentricity characterizes the shape of the ellipse. Indeed, from formula (28) it follows that the smaller the eccentricity of the ellipse, the less its semi-minor axis b differs from the semi-major axis a, i.e., the less elongated the ellipse is (along the focal axis).

In the limiting case, the result is a circle of radius a: , or . At the same time, the foci of the ellipse seem to merge at one point - the center of the circle. The eccentricity of the circle is zero:

The connection between the ellipse and the circle can be established from another point of view. Let us show that an ellipse with semi-axes a and b can be considered as a projection of a circle of radius a.

Let us consider two planes P and Q, forming between themselves such an angle a, for which (Fig. 46). Let us construct a coordinate system in the plane P, and in the plane Q - a system Oxy with common beginning coordinates O and a common abscissa axis coinciding with the line of intersection of the planes. Consider a circle in the plane P

with center at the origin and radius equal to a. Let be an arbitrarily chosen point on the circle, be its projection onto the Q plane, and let be the projection of point M onto the Ox axis. Let us show that the point lies on an ellipse with semi-axes a and b.

An ellipse is the geometric locus of points on a plane, the sum of the distances from each of them to two given points F_1, and F_2 is a constant value (2a) greater than the distance (2c) between these given points(Fig. 3.36, a). This geometric definition expresses focal property of an ellipse.

Focal property of an ellipse

Points F_1 and F_2 are called the foci of the ellipse, the distance between them 2c=F_1F_2 is the focal length, the middle O of the segment F_1F_2 is the center of the ellipse, the number 2a is the length of the major axis of the ellipse (accordingly, the number a is the semi-major axis of the ellipse). The segments F_1M and F_2M connecting an arbitrary point M of the ellipse with its foci are called focal radii of point M. The segment connecting two points of an ellipse is called a chord of the ellipse.

The ratio e=\frac(c)(a) is called the eccentricity of the ellipse. From definition (2a>2c) it follows that 0\leqslant e<1 . При e=0 , т.е. при c=0 , фокусы F_1 и F_2 , а также центр O совпадают, и эллипс является окружностью радиуса a (рис.3.36,6).

Geometric definition of ellipse, expressing its focal property, is equivalent to its analytical definition - the line given by the canonical equation of the ellipse:

Indeed, let us introduce a rectangular coordinate system (Fig. 3.36c). We take the center O of the ellipse as the origin of the coordinate system; we take the straight line passing through the foci (focal axis or first axis of the ellipse) as the abscissa axis (the positive direction on it is from point F_1 to point F_2); let us take a straight line perpendicular to the focal axis and passing through the center of the ellipse (the second axis of the ellipse) as the ordinate axis (the direction on the ordinate axis is chosen so that rectangular system coordinates Oxy turned out to be right).

Let's create an equation for the ellipse using its geometric definition, which expresses the focal property. In the selected coordinate system, we determine the coordinates of the foci F_1(-c,0),~F_2(c,0). For an arbitrary point M(x,y) belonging to the ellipse, we have:

\vline\,\overrightarrow(F_1M)\,\vline\,+\vline\,\overrightarrow(F_2M)\,\vline\,=2a.

Writing this equality in coordinate form, we get:

\sqrt((x+c)^2+y^2)+\sqrt((x-c)^2+y^2)=2a.

We move the second radical to the right side, square both sides of the equation and bring similar terms:

(x+c)^2+y^2=4a^2-4a\sqrt((x-c)^2+y^2)+(x-c)^2+y^2~\Leftrightarrow ~4a\sqrt((x-c )^2+y^2)=4a^2-4cx.

Dividing by 4, we square both sides of the equation:

a^2(x-c)^2+a^2y^2=a^4-2a^2cx+c^2x^2~\Leftrightarrow~ (a^2-c^2)^2x^2+a^2y^ 2=a^2(a^2-c^2).

Having designated b=\sqrt(a^2-c^2)>0, we get b^2x^2+a^2y^2=a^2b^2. Dividing both sides by a^2b^2\ne0, we arrive at the canonical equation of the ellipse:

\frac(x^2)(a^2)+\frac(y^2)(b^2)=1.

Therefore, the chosen coordinate system is canonical.

If the foci of the ellipse coincide, then the ellipse is a circle (Fig. 3.36,6), since a=b. In this case, any rectangular coordinate system with origin at the point will be canonical O\equiv F_1\equiv F_2, and the equation x^2+y^2=a^2 is the equation of a circle with center at point O and radius equal to a.

Carrying out the reasoning in reverse order, it can be shown that all points whose coordinates satisfy equation (3.49), and only they, belong to the locus of points called an ellipse. In other words, the analytical definition of an ellipse is equivalent to its geometric definition, which expresses the focal property of the ellipse.

Directorial property of an ellipse

The directrixes of an ellipse are two straight lines running parallel to the ordinate axis of the canonical coordinate system at the same distance \frac(a^2)(c) from it. At c=0, when the ellipse is a circle, there are no directrixes (we can assume that the directrixes are at infinity).

Ellipse with eccentricity 0 the locus of points in the plane, for each of which the ratio of the distance to a given point F (focus) to the distance to a given straight line d (directrix) not passing through a given point is constant and equal to the eccentricity e ( directorial property of an ellipse). Here F and d are one of the foci of the ellipse and one of its directrixes, located on one side of the ordinate axis of the canonical coordinate system, i.e. F_1,d_1 or F_2,d_2 .

In fact, for example, for focus F_2 and directrix d_2 (Fig. 3.37,6) the condition \frac(r_2)(\rho_2)=e can be written in coordinate form:

\sqrt((x-c)^2+y^2)=e\cdot\!\left(\frac(a^2)(c)-x\right)

Getting rid of irrationality and replacing e=\frac(c)(a),~a^2-c^2=b^2, we arrive at the canonical ellipse equation (3.49). Similar reasoning can be carried out for focus F_1 and director d_1\colon\frac(r_1)(\rho_1)=e.

Equation of an ellipse in a polar coordinate system

The equation of the ellipse in the polar coordinate system F_1r\varphi (Fig. 3.37, c and 3.37 (2)) has the form

r=\frac(p)(1-e\cdot\cos\varphi)

where p=\frac(b^2)(a) is the focal parameter of the ellipse.

In fact, let us choose the left focus F_1 of the ellipse as the pole of the polar coordinate system, and the ray F_1F_2 as the polar axis (Fig. 3.37, c). Then for an arbitrary point M(r,\varphi), according to the geometric definition (focal property) of an ellipse, we have r+MF_2=2a. We express the distance between points M(r,\varphi) and F_2(2c,0) (see):

\begin(aligned)F_2M&=\sqrt((2c)^2+r^2-2\cdot(2c)\cdot r\cos(\varphi-0))=\\ &=\sqrt(r^2- 4\cdot c\cdot r\cdot\cos\varphi+4\cdot c^2).\end(aligned)

Therefore, in coordinate form, the equation of the ellipse F_1M+F_2M=2a has the form

r+\sqrt(r^2-4\cdot c\cdot r\cdot\cos\varphi+4\cdot c^2)=2\cdot a.

We isolate the radical, square both sides of the equation, divide by 4 and present similar terms:

r^2-4\cdot c\cdot r\cdot\cos\varphi+4\cdot c^2~\Leftrightarrow~a\cdot\!\left(1-\frac(c)(a)\cdot\cos \varphi\right)\!\cdot r=a^2-c^2.

Express the polar radius r and make the replacement e=\frac(c)(a),~b^2=a^2-c^2,~p=\frac(b^2)(a):

r=\frac(a^2-c^2)(a\cdot(1-e\cdot\cos\varphi)) \quad \Leftrightarrow \quad r=\frac(b^2)(a\cdot(1 -e\cdot\cos\varphi)) \quad \Leftrightarrow \quad r=\frac(p)(1-e\cdot\cos\varphi),

Q.E.D.

Geometric meaning of the coefficients in the ellipse equation

Let's find the intersection points of the ellipse (see Fig. 3.37a) with the coordinate axes (vertices of the ellipse). Substituting y=0 into the equation, we find the points of intersection of the ellipse with the abscissa axis (with the focal axis): x=\pm a. Therefore, the length of the segment of the focal axis contained inside the ellipse is equal to 2a. This segment, as noted above, is called the major axis of the ellipse, and the number a is the semi-major axis of the ellipse. Substituting x=0, we get y=\pm b. Therefore, the length of the segment of the second axis of the ellipse contained inside the ellipse is equal to 2b. This segment is called the minor axis of the ellipse, and the number b is the semiminor axis of the ellipse.

Really, b=\sqrt(a^2-c^2)\leqslant\sqrt(a^2)=a, and the equality b=a is obtained only in the case c=0, when the ellipse is a circle. Attitude k=\frac(b)(a)\leqslant1 is called the ellipse compression ratio.

Notes 3.9

1. The straight lines x=\pm a,~y=\pm b limit the main rectangle on the coordinate plane, inside of which there is an ellipse (see Fig. 3.37, a).

2. An ellipse can be defined as the locus of points obtained by compressing a circle to its diameter.

Indeed, let the equation of a circle in the rectangular coordinate system Oxy be x^2+y^2=a^2. When compressed to the x-axis with a coefficient of 0

\begin(cases)x"=x,\\y"=k\cdot y.\end(cases)

Substituting circles x=x" and y=\frac(1)(k)y" into the equation, we obtain the equation for the coordinates of the image M"(x",y") of the point M(x,y) :

(x")^2+(\left(\frac(1)(k)\cdot y"\right)\^2=a^2 \quad \Leftrightarrow \quad \frac{(x")^2}{a^2}+\frac{(y")^2}{k^2\cdot a^2}=1 \quad \Leftrightarrow \quad \frac{(x")^2}{a^2}+\frac{(y")^2}{b^2}=1, !}

since b=k\cdot a . This is the canonical equation of the ellipse.

3. The coordinate axes (of the canonical coordinate system) are the axes of symmetry of the ellipse (called the main axes of the ellipse), and its center is the center of symmetry.

Indeed, if the point M(x,y) belongs to the ellipse . then the points M"(x,-y) and M""(-x,y), symmetrical to the point M relative to the coordinate axes, also belong to the same ellipse.

4. From the equation of the ellipse in the polar coordinate system r=\frac(p)(1-e\cos\varphi)(see Fig. 3.37, c), the geometric meaning of the focal parameter is clarified - this is half the length of the chord of the ellipse passing through its focus perpendicular to the focal axis (r=p at \varphi=\frac(\pi)(2)).

5. Eccentricity e characterizes the shape of the ellipse, namely the difference between the ellipse and the circle. The larger e, the more elongated the ellipse, and the closer e is to zero, the closer the ellipse is to a circle (Fig. 3.38a). Indeed, taking into account that e=\frac(c)(a) and c^2=a^2-b^2 , we get

e^2=\frac(c^2)(a^2)=\frac(a^2-b^2)(a^2)=1-(\left(\frac(a)(b)\right )\^2=1-k^2, !}

where k is the ellipse compression ratio, 0

6. Equation \frac(x^2)(a^2)+\frac(y^2)(b^2)=1 at a

7. Equation \frac((x-x_0)^2)(a^2)+\frac((y-y_0)^2)(b^2)=1,~a\geqslant b defines an ellipse with center at point O"(x_0,y_0), the axes of which are parallel to the coordinate axes (Fig. 3.38, c). This equation is reduced to the canonical one using parallel translation (3.36).

When a=b=R the equation (x-x_0)^2+(y-y_0)^2=R^2 describes a circle of radius R with center at point O"(x_0,y_0) .

Parametric equation of ellipse

Parametric equation of ellipse in the canonical coordinate system has the form

\begin(cases)x=a\cdot\cos(t),\\ y=b\cdot\sin(t),\end(cases)0\leqslant t<2\pi.

Indeed, substituting these expressions into equation (3.49), we arrive at the main trigonometric identity \cos^2t+\sin^2t=1.

Example 3.20. Draw an ellipse \frac(x^2)(2^2)+\frac(y^2)(1^2)=1 in the canonical coordinate system Oxy. Find the semi-axes, focal length, eccentricity, compression ratio, focal parameter, directrix equations.

Solution. Comparing the given equation with the canonical one, we determine the semi-axes: a=2 - semi-major axis, b=1 - semi-minor axis of the ellipse. We build the main rectangle with sides 2a=4,~2b=2 with the center at the origin (Fig. 3.39). Considering the symmetry of the ellipse, we fit it into the main rectangle. If necessary, determine the coordinates of some points of the ellipse. For example, substituting x=1 into the equation of the ellipse, we get

\frac(1^2)(2^2)+\frac(y^2)(1^2)=1 \quad \Leftrightarrow \quad y^2=\frac(3)(4) \quad \Leftrightarrow \ quad y=\pm\frac(\sqrt(3))(2).

Therefore, points with coordinates \left(1;\,\frac(\sqrt(3))(2)\right)\!,~\left(1;\,-\frac(\sqrt(3))(2)\right)- belong to the ellipse.

Calculating the compression ratio k=\frac(b)(a)=\frac(1)(2); focal length 2c=2\sqrt(a^2-b^2)=2\sqrt(2^2-1^2)=2\sqrt(3); eccentricity e=\frac(c)(a)=\frac(\sqrt(3))(2); focal parameter p=\frac(b^2)(a)=\frac(1^2)(2)=\frac(1)(2). We compose the directrix equations: x=\pm\frac(a^2)(c)~\Leftrightarrow~x=\pm\frac(4)(\sqrt(3)).

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 , as well as their lengths 2 (a > , as well as their lengths 2) - 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 ( , as well as their lengths 2, ABOUT) And (- , as well as their lengths 2, 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 = , as well as their lengths 2, 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/, as well as their lengths 2 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 , as well as their lengths 2= 4 . We obtain the canonical equation of the ellipse:

Points and , indicated in green on the major axis, where

B 1 tricks.

called eccentricity are called large and small respectively

Attitude , as well as their lengths 2/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 an 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 half of it, 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 foci 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

B 1 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.