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

Parametric equation of an ellipse

Parametric equation of an 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 basic trigonometric identity \cos^2t+\sin^2t=1 .

Example 3.20. draw ellipse \frac(x^2)(2^2)+\frac(y^2)(1^2)=1 in the canonical coordinate system Oxy . Find semiaxes, focal length, eccentricity, aspect ratio, focal parameter, directrix equations.

Decision. Comparing the given equation with the canonical one, we determine the semiaxes: a=2 - the major semiaxis, b=1 - the minor semiaxis of the ellipse. We build the main rectangle with sides 2a=4,~2b=2 centered at the origin (Fig.3.39). Given the symmetry of the ellipse, we fit it into the main rectangle. If necessary, we determine the coordinates of some points of the ellipse. For example, substituting x=1 into the ellipse equation, 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 an ellipse.

Calculate 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)).

Definition 7.1. The set of all points on the plane for which the sum of the distances to two fixed points F 1 and F 2 is a given constant is called ellipse.

The definition of an ellipse gives the following way of constructing it geometrically. We fix two points F 1 and F 2 on the plane, and denote a non-negative constant value by 2a. Let the distance between points F 1 and F 2 be equal to 2c. Imagine that an inextensible thread of length 2a is fixed at points F 1 and F 2, for example, with the help of two needles. It is clear that this is possible only for a ≥ c. Pulling the thread with a pencil, draw a line, which will be an ellipse (Fig. 7.1).

So, the described set is not empty if a ≥ c. When a = c, the ellipse is a segment with ends F 1 and F 2, and when c = 0, i.e. if the fixed points specified in the definition of an ellipse coincide, it is a circle of radius a. Discarding these degenerate cases, we will further assume, as a rule, that a > c > 0.

The fixed points F 1 and F 2 in definition 7.1 of the ellipse (see Fig. 7.1) are called ellipse tricks, the distance between them, denoted by 2c, - focal length, and the segments F 1 M and F 2 M, connecting an arbitrary point M on the ellipse with its foci, - focal radii.

The form of the ellipse is completely determined by the focal length |F 1 F 2 | = 2с and parameter a, and its position on the plane - by a pair of points F 1 and F 2 .

It follows from the definition of an ellipse that it is symmetrical about a straight line passing through the foci F 1 and F 2, as well as about a straight line that divides the segment F 1 F 2 in half and is perpendicular to it (Fig. 7.2, a). These lines are called ellipse axes. The point O of their intersection is the center of symmetry of the ellipse, and it is called the center of the ellipse, and the points of intersection of the ellipse with the axes of symmetry (points A, B, C and D in Fig. 7.2, a) - the vertices of the ellipse.

The number a is called semi-major axis of an ellipse, and b = √ (a 2 - c 2) - its semi-minor axis. It is easy to see that for c > 0, the major semiaxis a is equal to the distance from the center of the ellipse to those of its vertices that are on the same axis as the foci of the ellipse (vertices A and B in Fig. 7.2, a), and the minor semiaxis b is equal to the distance from the center ellipse to its other two vertices (vertices C and D in Fig. 7.2, a).

Ellipse equation. Consider some ellipse on the plane with foci at the points F 1 and F 2 , major axis 2a. Let 2c be the focal length, 2c = |F 1 F 2 |

We choose a rectangular coordinate system Oxy on the plane so that its origin coincides with the center of the ellipse, and the foci are on abscissa(Fig. 7.2, b). This coordinate system is called canonical for the ellipse under consideration, and the corresponding variables are canonical.

In the selected coordinate system, foci have coordinates F 1 (c; 0), F 2 (-c; 0). Using the formula for the distance between points, we write the condition |F 1 M| + |F 2 M| = 2a in coordinates:

√((x - c) 2 + y 2) + √((x + c) 2 + y 2) = 2a. (7.2)

This equation is inconvenient because it contains two square radicals. So let's transform it. We transfer the second radical in equation (7.2) to the right side and square it:

(x - c) 2 + y 2 = 4a 2 - 4a√((x + c) 2 + y 2) + (x + c) 2 + y 2 .

After opening the brackets and reducing like terms, we get

√((x + c) 2 + y 2) = a + εx

where ε = c/a. We repeat the squaring operation to remove the second radical: (x + c) 2 + y 2 = a 2 + 2εax + ε 2 x 2, or, given the value of the entered parameter ε, (a 2 - c 2) x 2 / a 2 + y 2 = a 2 - c 2 . Since a 2 - c 2 = b 2 > 0, then

x 2 /a 2 + y 2 /b 2 = 1, a > b > 0. (7.4)

Equation (7.4) is satisfied by the coordinates of all points lying on the ellipse. But when deriving this equation, nonequivalent transformations of the original equation (7.2) were used - two squarings that remove square radicals. Squaring an equation is an equivalent transformation if both sides contain quantities with the same sign, but we did not check this in our transformations.

We may not check the equivalence of transformations if we consider the following. A pair of points F 1 and F 2 , |F 1 F 2 | = 2c, on the plane defines a family of ellipses with foci at these points. Each point of the plane, except for the points of the segment F 1 F 2 , belongs to some ellipse of the specified family. In this case, no two ellipses intersect, since the sum of the focal radii uniquely determines a specific ellipse. So, the described family of ellipses without intersections covers the entire plane, except for the points of the segment F 1 F 2 . Consider a set of points whose coordinates satisfy equation (7.4) with a given value of the parameter a. Can this set be distributed among several ellipses? Some of the points of the set belong to an ellipse with a semi-major axis a. Let there be a point in this set lying on an ellipse with a semi-major axis a. Then the coordinates of this point obey the equation

those. equations (7.4) and (7.5) have common solutions. However, it is easy to verify that the system

for ã ≠ a has no solutions. To do this, it is enough to exclude, for example, x from the first equation:

which after transformations leads to the equation

having no solutions for ã ≠ a, because . So, (7.4) is the equation of an ellipse with the semi-major axis a > 0 and the minor semi-axis b = √ (a 2 - c 2) > 0. It is called the canonical equation of the ellipse.

Ellipse view. The geometric method of constructing an ellipse discussed above gives a sufficient idea of ​​​​the appearance of an ellipse. But the form of an ellipse can also be investigated with the help of its canonical equation (7.4). For example, considering y ≥ 0, you can express y in terms of x: y = b√(1 - x 2 /a 2), and, having examined this function, build its graph. There is another way to construct an ellipse. A circle of radius a centered at the origin of the canonical coordinate system of the ellipse (7.4) is described by the equation x 2 + y 2 = a 2 . If it is compressed with the coefficient a/b > 1 along y-axis, then you get a curve that is described by the equation x 2 + (ya / b) 2 \u003d a 2, i.e. an ellipse.

Remark 7.1. If the same circle is compressed with the coefficient a/b

Ellipse eccentricity. The ratio of the focal length of an ellipse to its major axis is called ellipse eccentricity and denoted by ε. For an ellipse given

canonical equation (7.4), ε = 2c/2a = с/a. If in (7.4) the parameters a and b are related by the inequality a

For c = 0, when the ellipse turns into a circle, and ε = 0. In other cases, 0

Equation (7.3) is equivalent to equation (7.4) because equations (7.4) and (7.2) are equivalent. Therefore, (7.3) is also an ellipse equation. In addition, relation (7.3) is interesting in that it gives a simple radical-free formula for the length |F 2 M| one of the focal radii of the point M(x; y) of the ellipse: |F 2 M| = a + εx.

A similar formula for the second focal radius can be obtained from symmetry considerations or by repeating calculations in which, before squaring equation (7.2), the first radical is transferred to the right side, and not the second. So, for any point M(x; y) on the ellipse (see Fig. 7.2)

|F 1 M | = a - εx, |F 2 M| = a + εx, (7.6)

and each of these equations is an ellipse equation.

Example 7.1. Let's find the canonical equation of an ellipse with semi-major axis 5 and eccentricity 0.8 and construct it.

Knowing the major semiaxis of the ellipse a = 5 and the eccentricity ε = 0.8, we find its minor semiaxis b. Since b \u003d √ (a 2 - c 2), and c \u003d εa \u003d 4, then b \u003d √ (5 2 - 4 2) \u003d 3. So the canonical equation has the form x 2 / 5 2 + y 2 / 3 2 \u003d 1. To construct an ellipse, it is convenient to draw a rectangle centered at the origin of the canonical coordinate system, the sides of which are parallel to the axes of symmetry of the ellipse and equal to its corresponding axes (Fig. 7.4). This rectangle intersects with

the axes of the ellipse at its vertices A(-5; 0), B(5; 0), C(0; -3), D(0; 3), and the ellipse itself is inscribed in it. On fig. 7.4 also shows the foci F 1.2 (±4; 0) of the ellipse.

Geometric properties of an ellipse. Let us rewrite the first equation in (7.6) as |F 1 M| = (а/ε - x)ε. Note that the value of a / ε - x for a > c is positive, since the focus F 1 does not belong to the ellipse. This value is the distance to the vertical line d: x = a/ε from the point M(x; y) to the left of this line. The ellipse equation can be written as

|F 1 M|/(а/ε - x) = ε

It means that this ellipse consists of those points M (x; y) of the plane for which the ratio of the length of the focal radius F 1 M to the distance to the straight line d is a constant value equal to ε (Fig. 7.5).

The line d has a "double" - a vertical line d", symmetrical to d with respect to the center of the ellipse, which is given by the equation x \u003d -a / ε. With respect to d", the ellipse is described in the same way as with respect to d. Both lines d and d" are called ellipse directrixes. The directrixes of the ellipse are perpendicular to the axis of symmetry of the ellipse, on which its foci are located, and are separated from the center of the ellipse by a distance a / ε \u003d a 2 / c (see Fig. 7.5).

The distance p from the directrix to the focus closest to it is called focal parameter of the ellipse. This parameter is equal to

p \u003d a / ε - c \u003d (a 2 - c 2) / c \u003d b 2 / c

The ellipse has another important geometric property: the focal radii F 1 M and F 2 M make equal angles with the tangent to the ellipse at the point M (Fig. 7.6).

This property has a clear physical meaning. If a light source is placed at the focus F 1, then the beam emerging from this focus, after reflection from the ellipse, will go along the second focal radius, since after reflection it will be at the same angle to the curve as before reflection. Thus, all the rays leaving the focus F 1 will be concentrated in the second focus F 2 and vice versa. Based on this interpretation, this property is called optical property of an ellipse.

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

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

The concept of an algebraic line and its order

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

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

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

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

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

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

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

For example:

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

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

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

Maximum value: 2

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

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

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

However, let us return to the general equation and recall its simplest school variations. Examples are the parabola, whose equation can be easily reduced to a general form, and the hyperbola with an equivalent equation. However, not everything is so smooth ....

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

What is the canonical form of an equation?

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

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

Classification of second order lines

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

( and are positive real numbers)

1) is the canonical equation of the ellipse;

2) is the canonical equation of the hyperbola;

3) is the canonical equation of the parabola;

4) – imaginary ellipse;

5) - a pair of intersecting lines;

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

7) - a pair of parallel lines;

8) - couple imaginary parallel lines;

9) is a pair of coinciding lines.

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

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

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

Ellipse and its canonical equation

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

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

How to build an ellipse?

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

Example 1

Construct an ellipse given by the equation

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

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

In this case :


Line segment called major axis ellipse;
line segment – minor axis;
number called semi-major axis ellipse;
number – semi-minor axis.
in our example: .

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

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

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

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

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

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

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

Mark points on the drawing (red color), symmetrical points on the other arcs (blue color) and carefully connect the whole company with a line:


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

Definition of an ellipse. Ellipse foci and ellipse eccentricity

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

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

Now it will become clearer:

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

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

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

How to find the focus of an ellipse?

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

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

Calculations are easier than steamed turnips:

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

The eccentricity of an ellipse and its geometric meaning

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

In our case:

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

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

Thus, the closer the eccentricity of the ellipse is to one, the more oblong the ellipse is.

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

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

A circle is a special case of an ellipse

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

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

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

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

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

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

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

Example 2

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

Solution and drawing at the end of the lesson

Let's add an action:

Rotate and translate an ellipse

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

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

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