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)).

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...

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 the generally accepted standard form of an 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 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 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 “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

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: .
In this case, the distances between the focuses are less than this 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 method 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 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 take this into account as you further explore 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 solution

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.

An ellipse is the geometric locus of points on a plane, the sum of the distances from each of which 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 the rectangular coordinate system Oxy is 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)).

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.

Let us establish the shape of the ellipse using its canonical 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 P plane, and in the Q plane a system Oxy with a common origin 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.

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 value is called ellipse.

The definition of an ellipse gives the following method of its geometric construction. 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 2c. Let's imagine that an inextensible thread of length 2a is fixed at points F 1 and F 2, for example, using two needles. It is clear that this is possible only for a ≥ c. Having pulled 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 foci, the distance between them, indicated 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 are focal radii.

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

From the definition of an ellipse it follows that it is symmetrical with respect to the line passing through the foci F 1 and F 2, as well as with respect to the 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) - vertices of the ellipse.

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

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

Let us choose a rectangular coordinate system Oxy on the plane so that its origin coincides with the center of the ellipse, and its foci are on x-axis(Fig. 7.2, b). Such a coordinate system is called canonical for the ellipse in question, and the corresponding variables are canonical.

In the selected coordinate system, the 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. Let us move 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 parentheses and bringing similar 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, taking into account 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 have quantities with the same sign, but we did not check this in our transformations.

We can avoid checking the equivalence of transformations if we take into account 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 indicated 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. Let us consider a set of points whose coordinates satisfy equation (7.4) with a given value of parameter a. Can this set be distributed among several ellipses? Some of the points of the set belong to an ellipse with semimajor axis a. Let there be a point in this set lying on an ellipse with semimajor 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

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

Ellipse view. The geometric method of constructing an ellipse discussed above gives a sufficient idea of ​​the appearance of the ellipse. But the shape of the ellipse can also be studied using its canonical equation (7.4). For example, you can, assuming y ≥ 0, express y through x: y = b√(1 - x 2 /a 2), and, having studied this function, build its graph. There is another way to construct an ellipse. A circle of radius a with center 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 a coefficient a/b > 1 along y-axis, then you get a curve that is described by the equation x 2 + (ya/b) 2 = a 2, i.e., an ellipse.

Remark 7.1. If the same circle is compressed by a factor a/b

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

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

When 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), since equations (7.4) and (7.2) are equivalent. Therefore, the equation of the ellipse is also (7.3). In addition, relation (7.3) is interesting because 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 equation of an ellipse.

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

Knowing the semi-major axis of the ellipse a = 5 and the eccentricity ε = 0.8, we will find its semi-minor axis b. Since b = √(a 2 - c 2), and c = εa = 4, then b = √(5 2 - 4 2) = 3. So the canonical equation has the form x 2 /5 2 + y 2 /3 2 = 1. To construct an ellipse, it is convenient to draw a rectangle with a center at the origin of the canonical coordinate system, the sides of which are parallel to the symmetry axes 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. In Fig. 7.4 also shows the foci F 1.2 (±4; 0) of the ellipse.

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

|F 1 M|/(a/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 straight line d has a “double” - the vertical straight line d, symmetrical to d relative to the center of the ellipse, which is given by the equation x = -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 directrixes of the ellipse. The directrixes of the ellipse are perpendicular to the axis of symmetry of the ellipse on which its foci are located, and are spaced from the center of the ellipse at a distance a/ε = 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 = a/ε - c = (a 2 - c 2)/c = 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 point M (Fig. 7.6).

This property has a clear physical meaning. If a light source is placed at focus F 1, then the ray 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 rays emerging from 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 the ellipse.