First order lines. Second order lines on the plane Canonical equation of a parabola

02.11.2023 | Essays

Second order curves on a plane are lines defined by equations in which the variable coordinates x And y are contained in the second degree. These include the ellipse, hyperbola and parabola.

The general form of the second order curve equation is as follows:

Where A, B, C, D, E, F- numbers and at least one of the coefficients A, B, C not equal to zero.

When solving problems with second-order curves, the canonical equations of the ellipse, hyperbola and parabola are most often considered. It is easy to move on to them from general equations; example 1 of problems with ellipses will be devoted to this.

Ellipse given by the canonical equation

Definition of an ellipse. An ellipse is the set of all points of the plane for which the sum of the distances to the points called foci is a constant value greater than the distance between the foci.

The focuses are indicated as in the figure below.

The canonical equation of an ellipse has the form:

Where a And b (a > b) - the lengths of the semi-axes, i.e. half the lengths of the segments cut off by the ellipse on the coordinate axes.

The straight line passing through the foci of the ellipse is its axis of symmetry. Another axis of symmetry of an ellipse is a straight line passing through the middle of a segment perpendicular to this segment. Dot ABOUT the intersection of these lines serves as the center of symmetry of the ellipse or simply the center of the ellipse.

The abscissa axis of the ellipse intersects at the points ( a, ABOUT) And (- a, ABOUT), and the ordinate axis is in points ( b, ABOUT) And (- b, ABOUT). These four points are called the vertices of the ellipse. The segment between the vertices of the ellipse on the x-axis is called its major axis, and on the ordinate axis - its minor axis. Their segments from the top to the center of the ellipse are called semi-axes.

If a = b, then the equation of the ellipse takes the form . This is the equation of a circle with radius a, and a circle is a special case of an ellipse. An ellipse can be obtained from a circle of radius a, if you compress it into a/b times along the axis Oy .

Example 1. Check if the line given is general equation , 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 resulting equation is canonical equation ellipse. Therefore, this line is an ellipse.

Example 2. Compose the canonical equation of an ellipse if its semi-axes are 5 and 4, respectively.

Solution. We look at the formula for the canonical equation of an ellipse and substitute: the semimajor axis is a= 5, the semiminor axis is b= 4 . We obtain the canonical equation of the ellipse:

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

are called tricks.

called eccentricity ellipse.

Attitude b/a characterizes the “oblateness” of the ellipse. The smaller this ratio, the more the ellipse is elongated along the major axis. However, the degree of elongation of an ellipse is more often expressed through eccentricity, the formula for which is given above. For different ellipses, the eccentricity varies from 0 to 1, always remaining less than unity.

Example 3. Compose the canonical equation of the ellipse if the distance between the foci is 8 and the major axis is 10.

Solution. Let's make some simple conclusions:

If the major axis is equal to 10, then its half, i.e. the semi-axis a = 5 ,

If the distance between the foci is 8, then the number c of the focal coordinates is equal to 4.

We substitute and calculate:

The result is the canonical equation of the ellipse:

Example 4. Compose the canonical equation of an ellipse if its major axis is 26 and its eccentricity is .

Solution. As follows from both the size of the major axis and the eccentricity equation, the semimajor axis of the ellipse a= 13. From the eccentricity equation we express the number c, needed to calculate the length of the minor semi-axis:

We calculate the square of the length of the minor semi-axis:

We compose the canonical equation of the ellipse:

Example 5. Determine the foci of the ellipse given by the canonical equation.

Solution. Find the number c, which determines the first coordinates of the ellipse's foci:

We get the focuses of the ellipse:

Example 6. The foci of the ellipse are located on the axis Ox symmetrical 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

are called headmistresses ellipse (in the drawing there are red lines along the edges).

From the two equations above it follows that for any point of the ellipse

where and are the distances of this point to the directrixes and .

Example 7. Given an ellipse. Write an equation for its directrixes.

Solution. We look at the directrix equation and find that we need to find the eccentricity of the ellipse, i.e. We have all the data for this. We calculate:

We obtain the equation of the directrixes of the ellipse:

Example 8. Compose the canonical equation of an ellipse if its foci are points and directrixes are lines.

(MIF-2, No. 3, 2005)

Second order lines on a plane

P. 1. Definition of a second order line

Consider a plane on which a rectangular Cartesian system coordinates (XOY). Then any point M is uniquely determined by its coordinates (x, y). In addition, any pair of numbers (x, y) defines some point on the plane. The coordinates of points may satisfy certain conditions, for example, some equation f(x, y) = 0 with respect to the unknowns (x, y). In this case, they say that the equation f(x, y)=0 defines a certain figure on the plane. Let's look at examples.

Example 1. Consider the function y= f( x). The coordinates of the points on the graph of this function satisfy the equation y– f( x) = 0.

Example 2. Equation (*), where a, b, c– some numbers define a certain straight line on the plane. (Equations of the form (*) are called linear).

Example 3. The graph of a hyperbola consists of points whose coordinates satisfy the equation https://pandia.ru/text/80/134/images/image004_92.gif" width="161" height="25">.

Definition 1. Equation of the form (**), where at least one of the coefficients is DIV_ADBLOCK53">

We will look at geometric and physical properties the lines mentioned above. Let's start with an ellipse.

https://pandia.ru/text/80/134/images/image008_54.gif" width="79" height="44 src="> (1).

Equation (1) is called canonical equation of an ellipse.

The shape of the ellipse can be judged from Figure 1.

Let's put it. The points are called tricks ellipse. There are a number of interesting properties associated with tricks, which we will discuss below.

Definition 4. Hyperbole is a figure on a plane whose coordinates of all points satisfy the equation

(2).

Equation (2) is called canonical hyperbola equation. The type of hyperbola can be judged from Figure 2.

Let's put it. The points are called tricks hyperbole. Parameter a called valid, and the parameter b- imaginary semi-axis hyperbolas, respectively ox– real, and oh– imaginary axis of the hyperbola.

https://pandia.ru/text/80/134/images/image016_34.gif" width="61" height="41">, are called asymptotes. For large parameter values x the points of the asymptotes approach the branches of the hyperbola infinitely close. In Figure 2, the asymptotes are depicted by dotted lines.

Definition 5. A parabola is a figure on a plane whose coordinates of all points satisfy the equation

https://pandia.ru/text/80/134/images/image018_28.gif" width="47" height="45 src=">.

P. 3. Properties of LVP focuses

For each LVP in A.2. special points were indicated - tricks. These points play a big role in explaining important properties ellipse, hyperbola and parabola. We formulate these properties in the form of theorems.

Theorem. 1. An ellipse is a set of pointsM, such that the sum of the distances from these points to the foci is equal to 2a:

https://pandia.ru/text/80/134/images/image020_26.gif" width="115" height="23 src="> (5).

In order to formulate a similar property for a parabola, we define headmistress. It's straight d, given by the equation https://pandia.ru/text/80/134/images/image022_23.gif" width="103" height="21 src="> (6).

P. 4. Focuses and tangents

https://pandia.ru/text/80/134/images/image024_24.gif" align="right" width="322" height="386 src=">.gif" width="52" height="24 src="> belongs to the corresponding HDL. Below are the equations for the tangents passing through this point:

– for an ellipse, (7)

– for hyperbole, (8)

- for a parabola. (9)

If we draw segments from both foci to the point of tangency with an ellipse or hyperbola (they are called focal radii points), then something remarkable will be revealed property(see Fig. 5 and 6): focal radii form equal angles with the tangent drawn at this point.

This property has an interesting physical interpretation. For example, if we consider the contour of an ellipse to be mirrored, then rays of light from a point source placed at one focus, after reflection from the walls of the circuit, will necessarily pass through the second focus.

Big practical use obtained a similar property for a parabola. The fact is that the focal radius of any point of the parabola makes an angle with the tangent drawn to this point, equal to angle between the tangent and the axis of the parabola.

Physically this is interpreted as follows: the rays of a point placed at the focus of the parabola, after reflection from its walls, propagate parallel to the axis of symmetry of the parabola. That is why the mirrors of lanterns and spotlights have a parabolic shape. By the way, if a stream of light (radio waves) parallel to the axis of the parabola enters it, then, after reflection from the walls, all its rays will pass through the focus. Space communications stations, as well as radars, operate on this principle.

P. 5. A little more physics

HDLs have found widespread use in physics and astronomy. Thus, it was found that one relatively light body (for example, a satellite) moves in the gravitational field of a more massive body (planet or star) along a trajectory that represents one of the LVPs. In this case, the more massive body is at the focus of this trajectory.

For the first time these properties were studied in detail Johannes Kepler and they were called Kepler's Laws.

Test No. 1 for 10th grade students

Self-test questions (5 points per task)

M.10.1.1. Define HDL. Give some examples of equations that define the LVP.

M.10.1.2. Calculate the coordinates of the foci of a) an ellipse, b) a hyperbola, if a=13, b=5.

M.10.1.3. Compose the canonical equation of a) an ellipse, b) a hyperbola, if it is known that this line passes through points with coordinates (5, 6) and (-8, 7).

M.10.1.4. Check that the straight line given by equation (9) actually intersects the parabola given by equation (3) only at the point with coordinates . ( Note: first substitute the equation of the tangent into the equation of the parabola, and then make sure that the discriminant of the resulting quadratic equation equal to zero.)

M.10.1.5. Write an equation for the tangent to the hyperbola with real semi-axis 8 and imaginary semi-axis – 4 at the point with coordinate x=11 if the second coordinate of the point is negative.

Practical work (10 points)

M.10.1.6. Construct several ellipses using the following method: secure a sheet of paper to plywood and stick a couple of buttons into the paper (but not all the way). Take a piece of thread and tie the ends. Throw the resulting loop over both buttons (the focal points of the future ellipse), pull the thread with the sharp end of a pencil and carefully draw a line, making sure that the thread is taut. By changing the dimensions of the loop, you can build several confocal ellipses. Try to explain using Theorem 1 that the resulting lines are really ellipses and explain how, knowing the distance between the buttons and the length of the thread, you can calculate the semi-axes of the ellipse.

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

Equation coefficients real numbers, but at least one of the numbers A, B or C is different from 0. such lines are called lines (curves) of the second order. Below we will show that equation (1) defines an Ellipse, a hyperbola or a parabola on a plane.

Circle

The simplest second-order curve is a circle. Recall that a circle of radius R with center at point M 0 is called the set of points M of the plane satisfying the condition MM 0 =R. Let the point M 0 in the Oxy system have coordinates x 0 ,y 0 , and M(x,y) be an arbitrary point on the circle. Then or

-canonical equation of a circle . Assuming x 0 =y 0 =0 we obtain x 2 +y 2 =R 2

Let us show that the equation of a circle can be written as a general equation of the second degree (1). To do this, we square the right side of the circle equation and get:

In order for this equation to correspond to (1) it is necessary that:

1) coefficient B=0,

2) . Then we get: (2)

The last equation is called general equation of a circle . Dividing both sides of the equation by A ≠0 and adding the terms containing x and y to a complete square, we get:

(2)

Comparing this equation with the canonical equation of a circle, we find that equation (2) is truly an equation of a circle if:

1)A=C, 2)B=0, 3)D 2 +E 2 -4AF>0.

If these conditions are met, the center of the circle is located at point O, and its radius .

Ellipse

F 2 (c,o)

F 1 (-c,o)

By definition 2 >2c, that is, >c. To derive the equation of the ellipse, we will assume that the foci F 1 and F 2 lie on the Ox axis, and t.O coincides with the middle of the segment F 1 F 2, then F 1 (-c, 0), F 2 (c,0).

Let M(x,y) be an arbitrary point of the ellipse, then, according to the definition of the ellipse MF 1 +MF 2 =2 that is

This is the equation of an ellipse. You can convert it to a simpler form as follows:

Square it:

square it

Since 2 -c 2 >0 we put 2 -c 2 =b 2

Then the last equation will take the form:

is the equation of an ellipse in canonical form.

The shape of the ellipse depends on the ratio: when b= the ellipse turns into a circle. The equation will take the form . The ratio is often used as a characteristic of an ellipse. This quantity is called the eccentricity of the ellipse, and 0< <1 так как 0

Study of the shape of an ellipse.

1) the equation of the ellipse contains x and y, only to an even degree, therefore the ellipse is symmetrical with respect to the axes Ox and Oy, as well as with respect to TO (0,0), which is called the center of the ellipse.

2) find the points of intersection of the ellipse with the coordinate axes. Setting y=0 we find A 1 ( ,0) and A 2 (- ,0), in which the ellipse intersects Ox. Putting x=0, we find B 1 (0,b) and B 2 (0,-b). Points A 1 , A 2 , B 1 , B 2 are called the vertices of the ellipse. The segments A 1 A 2 and B 1 B 2, as well as their lengths 2 and 2b, are called the major and minor axes of the ellipse, respectively. The numbers and b are the major and minor semi-axes, respectively.

A 1 ( ,0)

A2(- ,0)

B 2 (0,b)

Consequently, all points of the ellipse lie inside the rectangle formed by the lines x=± ,y=±b. (Fig.2.)

4) In the ellipse equation, the sum of non-negative terms is equal to one. Consequently, as one term increases, the other will decrease, that is, if |x| increases, then |y| - decreases and vice versa. From all that has been said, it follows that the ellipse has the shape shown in Fig. 2. (oval closed curve).

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 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'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 on 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.

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Lecture No. 9. Topic 3: Second order lines

Let a line defined by a second-degree equation be given in some DSC

where are the coefficients
are not equal to zero at the same time. This line is called curve or second order line.

It may happen that there are no points
with real coordinates satisfying equation (1). In this case, it is believed that equation (1) defines an imaginary line of the second order. For example,
 This is the equation of an imaginary circle.

Let us consider three important special cases of equation (1).

3.1. Ellipse

The ellipse is defined by the equation

(2)

Odds A And b are called the semi-major and semi-minor axes, respectively, and equation (2) is canonical equation of an ellipse.

Let's put
and mark on the axis ABOUT Xpoints

called tricks ellipse. Then the ellipse can be defined as

locus of points, the sum of the distances from which to the foci is a constant value equal to 2A.

M K

— AF 1 O F 2 a x

— b

Let's show it. Let the point
 current point of the ellipse. In this case we get Then the equality must hold

Let us represent expression (3) in the form

and square both sides of the expression

From here we get

Once again, let's square this expression and use the relationship
, Then

(4)

Dividing both sides of expression (4) by
, we finally obtain the canonical equation of the ellipse

Let us examine equation (2). If we replace , then equation (2) will not change. This means that the ellipse is symmetrical about the coordinate axes. Therefore, let us consider in detail the part of the ellipse located in the first quarter. It is determined by the equation
It is obvious that the ellipse passes through the points
. Having completed the schematic construction in the first quarter, we will symmetrically display its graph in all quarters. Thus, the ellipse is a continuous closed curve. The points are called vertices of the ellipse.

Attitude
calledeccentricityellipse. For ellipse
.

Direct
are called directrixes of the ellipse.

The following property of directrixes is true::

The ratio of the distances from the focus and the directrix for the points of the ellipse is a constant value equal to the eccentricity, i.e.

It is proved in the same way as equality (3).

Note 1. Circle
is a special case of an ellipse. For her

3.2. Hyperbola

The canonical equation of a hyperbola has the form

those. in equation (1) we need to put

Odds A And b are called the real and imaginary semi-axes, respectively.

Putting
, mark on the axis ABOUT Xpoints
called tricks hyperbole. Then a hyperbola can be defined as

locus of points, the difference in distances from which to the foci in absolute value is 2A, i.e.

TO M

F 1 —A ABOUT AF 2 X

The proof is similar to that for the ellipse. Based on the form of the hyperbola equation, we also conclude that its graph is symmetrical with respect to the axes of the coordinate system. The part of the hyperbola lying in the first quarter has the equation
From this equation it is clear that for sufficiently largeXhyperbola is close to a straight line
. After schematic construction in the first quarter, we symmetrically display the graph in all quarters.

Points
are called peaks hyperbole. Direct
are called asymptotes - these are the straight lines to which the branches of the hyperbola tend without intersecting them.

The relationship is calledeccentricityhyperbole. For hyperbole
.

Direct lines are called headmistresses hyperbole. For the directrixes of a hyperbola, a property similar to that for the directrixes of an ellipse holds.

Example. Find the equation of an ellipse whose vertices are at the foci, and the foci are at the vertices of the hyperbola
.