It can be shown (we do not) that equation (2) is equivalent to equation (1), although it is derived from (1) by nonequivalent transformations. This means that equation (2) is the equation of this ellipse. It's called canonical(i.e. the simplest).

It can be seen that the equation of the ellipse is a 2nd order equation, i.e. ellipse line of 2nd order.

For an ellipse we introduce the concept eccentricity. This is the quantity. For an ellipse, eccentricity is . Because With And A known, then also known. The expression for the focal radii of the point M(x, y) of the ellipse is easily obtained from the previous arguments: . r 2 will be found from equality (3)

Comment If you drive two nails (F1 and F2) into the table, tie a string to them at both ends, the length of which is greater than the distance between the nails ( 2a), pull the cord and draw a piece of chalk along the table, then it will draw a closed ellipse curve that is symmetrical about both axes and the origin.

4. Study of the shape of an ellipse using its canonical equation.

In the remark, for reasons of clarity, we concluded about the shape of the ellipse. Let us now study the shape of the ellipse by analyzing its canonical equation:

Let's find the points of intersection with the coordinate axes. If ,y=0, then , , i.e. we have two points A1(-a,0) and A2(a,0). If x=0, then , . Those. we have two points B1(0,-b) and B2(0,b) (since , then ). Points A1, A2, B1, B2 are called vertices of the ellipse.

2) The location area of ​​the ellipse can be determined from the following considerations:

a) from the equation of the ellipse it follows that, i.e. , i.e. or .

b) similarly, i.e. or . This shows that the entire ellipse is located in the rectangle formed by the lines and .

3) Further, the variables x and y enter the equation of the ellipse only in even powers, which means that the curve is symmetrical with respect to each of the axes and with respect to the origin. D-but, if a point (x, y) belongs to the radius, then the points (x, -y), (-x, y) and (-x, -y) also belong to it. Therefore, it is enough to consider only that part of the ellipse that lies in the first quarter, where and .

4) From the equation of the ellipse we have , and in the first quarter . If x=0, then y=b. This is point B2(0,b). Let x increase from 0 to a, then y decreases from b to 0. Thus, point M(x, y), starting from point B2(0, b) describing an arc, comes to point A(a,0). It can be strictly proven that the arc is convexly directed upward. By mirroring this arc in the coordinate axes and the origin, we get the entire ellipse. The axes of symmetry of an ellipse are called its axes; the point O of their intersection is the center of the ellipse. The length of the segments OA1=OA2=a is called the semi-major axis of the ellipse, the segments OB1, OB2=b are the semi-minor axis of the ellipse, (a>b), c is the half-focal distance. The magnitude is easy to explain geometrically.

When a=b we obtain from the canonical equation of the ellipse the equation of a circle. For a circle, i.e. F1=F2=0. .

Thus, a circle is a special case of an ellipse, when its foci coincide with the center and eccentricity = 0. The greater the eccentricity, the more elongated the ellipse.

Comment. From the canonical equation of the ellipse it is easy to conclude that the ellipse can be specified in parametric form. x=a cos t

y=b sin t, where a, b are the major and minor semi-axes, t-angle.

5. Definition and derivation of the canonical hyperbola equation.

Hyperbole called HMT planes, for which the difference in distances from two fixed points F1F2 of the plane, called foci, is a constant value (not equal to 0 and less than the focal length F1F2).

We will denote, as before, F1F2 = 2c, and the difference in distances is 2a (a<с). Систему координат выберем как и в случае эллипса.

Let M (x,y) be the current point of the hyperbola. By definition MF1-MF2= or r 1 -r 2 = = or --(1). – this is the equation of a hyperbola.

We get rid of irrationality in (1): we isolate one root, square both parts, we get: or , square it again:

Where .

Divide by . Let us introduce the designation . Then --(2). Equation (2), as can be shown, is equivalent to equation (1), and therefore is the equation of a given hyperbola. He is called the canonical equation of a hyperbola. We see that the hyperbola equation is also of the second degree, which means hyperbola line of second order.

Eccentricity of a hyperbola. The expression for focal radii through is easy to obtain from the previous one, then we find it from .

6. Study of the shape of a hyperbola using its canonical equation.

We reason in the same way as when studying an ellipse.

1. Find the points of intersection with the axes of the hyperbola. If x=0, then . There are no points of intersection with the op-amp axis. If y=0, then . Intersection points , . They're called vertices of the hyperbola.

2. Area of ​​location of the hyperbola: , i.e. or . This means that the hyperbola is located outside the strip bounded by straight lines x=-a And x=a.

3. Hyperbola has all types of symmetry, because x and y occur in even powers. Therefore, it is enough to consider that part of the hyperbola that is located in the first quarter.

4. From the hyperbola equation (2) in the first quarter we have . For x=a, y=0 we have the point ; at , i.e. the curve goes up to the right. To imagine the move more clearly, consider two auxiliary lines passing through the origin of coordinates and being the diagonals of a rectangle with sides 2a and 2b: BCB’C’. They have equations and . Let us prove that the current point of the hyperbola M(x,y) goes to infinity and approaches the straight line without limit. Let's take an arbitrary point X and compare the corresponding ordinates of the point of the hyperbola and the line. It's obvious that Y>y. MN=Y-y= .

We see that when , i.e. the curve indefinitely approaches the straight line as it moves away from the origin. This proves that the line is an asymptote of the hyperbola. Moreover, the hyperbola does not intersect the asymptote. This is enough to construct part of the hyperbola. It is convexly facing upward. The remaining parts are completed in symmetry. Note that the symmetry axes of a hyperbola (coordinate axes) are called its axes, the point of intersection of the axes- center hyperbole. One axis intersects the hyperbola (real axis), the other does not (imaginary). Line segment A called the real semi-axis, the segment b- imaginary semi-axis. The rectangle BCB'C' is called the basic rectangle of the hyperbola.

If a=b, then the asymptotes form angles with the coordinate axes along . Then the hyperbole is called equilateral or equilateral. The main rectangle turns into a square. Its asymptotes are perpendicular to each other.

Comment.

Sometimes we consider a hyperbola whose canonical equation is (3). They call her conjugate in relation to hyperbole (2). Hyperbola (3) has a real axis that is vertical and an imaginary axis that is horizontal. Its appearance is immediately established if you rearrange X And at, A And b(she turns back to her old self). But then hyperbola (3) has the form:

Its peaks.

5.As already indicated, the equation of an equilateral hyperbola ( a=b), when the coordinate axes coincide with the axes of the hyperbola, has the form . (4)

Because the asymptotes of an equilateral hyperbola are perpendicular, then they can also be taken as the coordinate axes OX 1 and OU 1. This is equivalent to turning the previous OXY system by an angle. The angle rotation formulas are as follows:

Then in the new coordinate system OX 1 Y 1 equation (4) will be rewritten:

Or or . Denoting , we get or (5) - this is the equation equilateral hyperbola, classified as asymptotes (it was this type of hyperbola that was considered at school).

Comment: From the equation it follows that the area of ​​any rectangle constructed on the coordinates of any point of the hyperbola M(x,y) is the same: S= k 2 .

7. Definition and derivation of the canonical equation of a parabola.

Parabola is called the GMT of the plane, for each of which the distance from a fixed point F of the plane, called focus, is equal to the distance from a fixed straight line called headmistress(focus outside the headmistress).

We will denote the distance from F to the directrix by p and call it the parameter of the parabola. Let us choose the coordinate system as follows: draw the OX axis through the point F perpendicular to the directrix NP. Let us choose the origin of coordinates in the middle of the segment FP.

In this system: .

Let's take an arbitrary point M(x,y) with current coordinates (x,y). That's why

Hence (1) is the equation of the parabola. Let's simplify:

Or (2) - this is it canonical equation of a parabola. It can be shown that (1) and (2) are equivalent.

Equation (2) is a 2nd order equation, i.e. parabola is a line of 2nd order.

8. Study of the shape of a parabola using its canonical equation.

(p>0).

1) x=0, y=0 the parabola passes through the origin of coordinates point O. It is called the vertex of the parabola.

2), i.e. the parabola is located to the right of the op-amp axis, in the right half-plane.

3) at is included in an even degree, therefore the parabola is symmetrical about the OX axis, therefore, it is enough to construct it in the first quarter.

4) in 1st quarter at , i.e. the parabola goes up to the right. It can be shown that the convexity is upward. We build at the bottom according to symmetry. The axis OU is tangent to the parabola.

Obviously, the focal radius is . The relationship is called eccentricity: . The axis of symmetry of a parabola (in our case OX) is called the axis of the parabola.

Note that the equation is also a parabola, but directed in the opposite direction. The equations also define parabolas, the axis of which is the axis of the op-amp.

or in a more familiar form, where .

The equation defines an ordinary parabola with a displaced vertex.

Notes. 1) There is a close relationship between all four lines of the 2nd order - they are all conic sections. If we take a cone of two cavities, then when we cut it with a plane perpendicular to the axis of the cone we get a circle; if we slightly tilt the section plane we get an ellipse; if the plane is parallel to the generatrix, then the section is a parabola, if the plane intersects both

cavities-hyperbola.

2) It can be proven that if a ray of light coming from the focus of a parabola is reflected from it, then the reflected ray goes parallel to the axis of the parabola - this is used in the action of spotlights - a parabolic reflector, and at the focus - a light source. This results in a directed stream of light.

3) If we imagine the launch of an Earth satellite from point T lying outside the atmosphere in the horizontal direction, then if the initial speed v 0 is insufficient, then the satellite will not rotate around the Earth. Upon reaching escape velocity 1, the satellite will rotate around the Earth in a circular orbit with its center at the center of the Earth. If the initial speed is increased, then the rotation will occur along an ellipse, the center of the Earth will be at one of the foci. Upon reaching the 2nd escape velocity, the trajectory will become parabolic and the satellite will not return to point T, but will be within the Solar System. Those. A parabola is an ellipse with one focus at infinity. With a further increase in the initial speed, the trajectory will become hyperbolic and a second focus will appear on the other side. The center of the Earth will always be at the focus of the orbit. The satellite will leave the solar system.

11.1. Basic Concepts

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

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

11.2. Circle

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

Then from the condition we obtain the equation

(11.2)

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

Equation (11.2) is called canonical equation of a circle

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

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

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

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

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

Let's transform this equation:

(11.4)

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

.

If , then equation (11.3) has the form

.

It is satisfied by the coordinates of a single point . In this case they say: “the circle has degenerated into a point” (has zero radius).

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

11.3. Ellipse

Canonical ellipse equation

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

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

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

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

This, in essence, is the equation of an ellipse.

Let us transform equation (11.5) to a simpler form as follows:

Because a>With, That . Let's put

(11.6)

Then the last equation will take the form or

(11.7)

It can be proven that equation (11.7) is equivalent to the original equation. It's called canonical ellipse equation .

An ellipse is a second order curve.

Study of the shape of an ellipse using its equation

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

1. Equation (11.7) contains x and y only in even powers, so if a point belongs to an ellipse, then the points ,, also belong to it. It follows that the ellipse is symmetrical with respect to the and axes, as well as with respect to the point, which is called the center of the ellipse.

2. Find the points of intersection of the ellipse with the coordinate axes. Putting , we find two points and , at which the axis intersects the ellipse (see Fig. 50). Putting in equation (11.7) , we find the points of intersection of the ellipse with the axis: and . Points A 1 , A 2 , B 1, B 2 are called vertices of the ellipse. Segments A 1 A 2 And B 1 B 2, as well as their lengths 2 a and 2 b are called accordingly major and minor axes ellipse. Numbers a And b are called large and small respectively axle shafts ellipse.

3. From equation (11.7) it follows that each term on the left side does not exceed one, i.e. the inequalities and or and take place. Consequently, all points of the ellipse lie inside the rectangle formed by the straight lines.

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

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

More information about the ellipse

The shape of the ellipse depends on the ratio. When the ellipse turns into a circle, the equation of the ellipse (11.7) takes the form . The ratio is often used to characterize the shape of an ellipse. The ratio of half the distance between the foci to the semi-major axis of the ellipse is called the eccentricity of the ellipse and o6o is denoted by the letter ε (“epsilon”):

with 0<ε< 1, так как 0<с<а. С учетом равенства (11.6) формулу (11.8) можно переписать в виде

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

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

The formulas hold

Direct lines are called

Theorem 11.1. If is the distance from an arbitrary point of the ellipse to some focus, d is the distance from the same point to the directrix corresponding to this focus, then the ratio is a constant value equal to the eccentricity of the ellipse:

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

11.4. Hyperbola

Canonical hyperbola equation

Hyperbole is the set of all points of the plane, the modulus of the difference in distances from each of them to two given points of this plane, called tricks , is a constant value less than the distance between the foci.

Let us denote the focuses by F 1 And F 2 the distance between them is 2s, and the modulus of the difference in distances from each point of the hyperbola to the foci through 2a. A-priory 2a < 2s, i.e. a < c.

To derive the hyperbola equation, we choose a coordinate system so that the foci F 1 And F 2 lay on the axis, and the origin coincided with the middle of the segment F 1 F 2(see Fig. 53). Then the foci will have coordinates and

Let be an arbitrary point of the hyperbola. Then, according to the definition of a hyperbola or , i.e. After simplifications, as was done when deriving the equation of the ellipse, we obtain canonical hyperbola equation

(11.9)

(11.10)

A hyperbola is a line of the second order.

Studying the shape of a hyperbola using its equation

Let us establish the form of the hyperbola using its caconical equation.

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

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

The points are called peaks hyperbolas, and the segment

real axis , line segment - real semi-axis hyperbole.

The segment connecting the points is called imaginary axis , number b - imaginary semi-axis . Rectangle with sides 2a And 2b called basic rectangle of hyperbola .

3. From equation (11.9) it follows that the minuend is not less than one, i.e., that or . This means that the points of the hyperbola are located to the right of the line (right branch of the hyperbola) and to the left of the line (left branch of the hyperbola).

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

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

Asymptotes of a hyperbola

The straight line L is called an asymptote unbounded curve K, if the distance d from point M of curve K to this straight line tends to zero when the distance of point M along curve K from the origin is unlimited. Figure 55 provides an illustration of the concept of an asymptote: straight line L is an asymptote for curve K.

Let us show that the hyperbola has two asymptotes:

(11.11)

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

Let us take a point N on a straight line that has the same abscissa x as the point on the hyperbola (see Fig. 56), and find the difference ΜΝ between the ordinates of the straight line and the branch of the hyperbola:

As you can see, as x increases, the denominator of the fraction increases; the numerator is a constant value. Therefore, the length of the segment ΜΝ tends to zero. Since MΝ is greater than the distance d from the point M to the line, then d tends to zero. So, the lines are asymptotes of the hyperbola (11.9).

When constructing a hyperbola (11.9), it is advisable to first construct the main rectangle of the hyperbola (see Fig. 57), draw straight lines passing through the opposite vertices of this rectangle - the asymptotes of the hyperbola and mark the vertices and , of the hyperbola.

Equation of an equilateral hyperbola.

the asymptotes of which are the coordinate axes

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

(11.12)

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

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

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

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

More information about hyperbole

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

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

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

The eccentricity of an equilateral hyperbola is . Really,

Focal radii And for points of the right branch the hyperbolas have the form and , and for the left branch - And .

Direct lines are called directrixes of a hyperbola. Since for a hyperbola ε > 1, then . This means that the right directrix is ​​located between the center and the right vertex of the hyperbola, the left - between the center and the left vertex.

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

The curve defined by the equation is also a hyperbola, the real axis 2b of which is located on the Oy axis, and the imaginary axis 2 a- on the Ox axis. In Figure 59 it is shown as a dotted line.

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

11.5. Parabola

Canonical parabola equation

A parabola is the set of all points of the plane, each of which is equally distant from a given point, called the focus, and a given line, called the directrix. The distance from the focus F to the directrix is ​​called the parameter of the parabola and is denoted by p (p > 0).

To derive the equation of the parabola, we choose the coordinate system Oxy so that the Ox axis passes through the focus F perpendicular to the directrix in the direction from the directrix to F, and the origin of coordinates O is located in the middle between the focus and the directrix (see Fig. 60). In the chosen system, the focus F has coordinates , and the directrix equation has the form , or .

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

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

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

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

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

It is easy to show that the graph of a quadratic trinomial, where , B and C are any real numbers, is a parabola in the sense of its definition given above.

11.6. General equation of second order lines

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

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

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

The equation

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

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

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

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

General second order equation

Let us now consider a general equation of the second degree with two unknowns:

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

Using axis rotation formulas

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

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

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

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

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

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 tasks. So, for example, according to the canonical equation "flat" straight, firstly, it is immediately clear that this is a straight line, and secondly, the point belonging to it and the direction vector are easily visible.

It is obvious that any 1st order line is a straight line. On the second floor, it is no longer the watchman who is waiting for us, but a much more diverse company of nine statues:

Classification of second order lines

Using a special set of actions, any equation of a second-order line is reduced to one of the following forms:

( and are positive real numbers)

1) – canonical equation of the ellipse;

2) – canonical equation of a hyperbola;

3) – canonical equation of a parabola;

4) – imaginary ellipse;

5) – a pair of intersecting lines;

6) – pair imaginary intersecting lines (with a single valid point of intersection at the origin);

7) – a pair of parallel lines;

8) – pair imaginary parallel lines;

9) – a pair of coincident lines.

Some readers may have the impression that the list is incomplete. For example, in point No. 7, the equation specifies the pair direct, parallel to the axis, and the question arises: where is the equation that determines the lines parallel to the ordinate axis? Answer: it not considered canonical. Straight lines represent the same standard case, rotated by 90 degrees, and the additional entry in the classification is redundant, since it does not bring anything fundamentally new.

Thus, there are nine and only nine 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 receive virtually no attention in the standard course of analytical geometry. And, in accordance with more current needs, we immediately move on to the strict definition of an ellipse:

Ellipse is the set of all points of the plane, the sum of the distances to each of which from two given points, called tricks ellipse, is a constant quantity, numerically equal to 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 sheet of paper, keeping the green thread tightly taut. Continue the process until you return to the starting point... great... the drawing can be checked by the doctor and teacher =)

How to find the foci of an ellipse?

In the above example, I depicted “ready-made” focal points, and now we will learn how to extract them from the depths of geometry.

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

The calculations are simpler than simple:

! The specific coordinates of foci cannot be identified with the meaning of “tse”! I repeat that this is DISTANCE from each focus to the center(which in the general case does not have to be located exactly at the origin).
And, therefore, the distance between the foci also cannot be tied to the canonical position of the ellipse. In other words, the ellipse can be moved to another place and the value will remain unchanged, while the foci will naturally change their coordinates. Please 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.

It is a geometric figure that is bounded by a curve given by the equation.

It has two focuses . Focuses are called such two points, the sum of the distances from which to any point of the ellipse is a constant value.

Ellipse figure drawing

F 1, F 2 – focuses. F 1 = (c; 0); F 2 (- c ; 0)

c – half the distance between focuses;

a – semimajor axis;

b – semi-minor axis.

Theorem.The focal length and semi-axes are related by the relation:

a 2 = b 2 + c 2 .

Proof: If point M is located at the intersection of the ellipse with the vertical axis, r 1 + r 2 = 2* (according to the Pythagorean theorem). If point M is located at its intersection with the horizontal axis, r 1 + r 2 = a – c + a + c. Because by definition, the sum r 1 + r 2 is a constant value, then, equating, we get:

r 1 + r 2 = 2 a.

Eccentricity of an ellipse figure

Definition. The shape of the ellipse is determined by the characteristic, which is the ratio of the focal length to the major axis and is called eccentricity.

Because With< a , то е < 1.

Definition. The quantity k = b / a is called compression ratio, and the quantity 1 – k = (a – b)/ a is called compression.

The compression ratio and eccentricity are related by the relation: k 2 = 1 – e 2 .

If a = b (c = 0, e = 0, the foci merge), then the ellipse turns into a circle.

If the condition is satisfied for the point M(x 1, y 1): then it is located inside the ellipse, and if , then the point is outside it.

Theorem.For an arbitrary point M(x, y) belonging to the ellipse figure, the following relations are true::

r 1 = a – ex, r 2 = a + ex.

Proof. It was shown above that r 1 + r 2 = 2 a. In addition, from geometric considerations we can write:

After squaring and bringing similar terms:

It is proved in a similar way that r 2 = a + ex. The theorem has been proven.

Directrix figures ellipse

The ellipse figure is associated with two straight lines called headmistresses. Their equations are:

x = a/e; x = - a / e .

Theorem.In order for a point to lie on the boundary of an ellipse figure, it is necessary and sufficient that the ratio of the distance to the focus to the distance to the corresponding directrix is ​​equal to the eccentricity e.

Example. Construct an ellipse passing through the left focus and the lower vertex of the figure, given by the equation:

Points F 1 (–c, 0) and F 2 (c, 0), where they are called ellipse foci , while the value is 2 c defines interfocal distance .

Points A 1 (–A, 0), A 2 (A, 0), IN 1 (0, –b), B 2 (0, b) are called vertices of the ellipse (Fig. 9.2), while A 1 A 2 = 2A forms the major axis of the ellipse, and IN 1 IN 2 – small, – the center of the ellipse.

The main parameters of the ellipse, characterizing its shape:

ε = With/a – ellipse eccentricity ;

– focal radii of the ellipse (dot M belongs to the ellipse), and r 1 = a + εx, r 2 = a – εx;

– directrixes of the ellipse .

For an ellipse it is true: the directrixes do not intersect the boundary and the internal region of the ellipse, and also have the property

The eccentricity of an ellipse expresses its degree of “compression”.

If b > a> 0, then the ellipse is given by equation (9.7), for which, instead of condition (9.8), the condition is satisfied

Then 2 A– minor axis, 2 b– major axis, – foci (Fig. 9.3). Wherein r 1 + r 2 = 2b,
ε = c/b, the directrixes are determined by the equations:

Given the condition we have (in the form of a special case of an ellipse) a circle of radius R = a. Wherein With= 0, which means ε = 0.

The points of the ellipse have characteristic property : the sum of the distances from each of them to the foci is a constant value equal to 2 A(Fig. 9.2).

For parametric definition of an ellipse (formula (9.7)) in cases where conditions (9.8) and (9.9) are met as a parameter t the angle between the radius vector of a point lying on the ellipse and the positive direction of the axis can be taken Ox:

If the center of an ellipse with semi-axes is at a point, then its equation has the form:

Example 1. Give the equation of the ellipse x 2 + 4y 2 = 16 to the canonical form and determine its parameters. Draw an ellipse.

Solution. Let's divide the equation x 2 + 4y 2 = 16 by 16, after which we get:

Based on the form of the resulting equation, we conclude that this is the canonical equation of an ellipse (formula (9.7)), where A= 4 – semimajor axis, b= 2 – semi-minor axis. This means that the vertices of the ellipse are the points A 1 (–4, 0), A 2 (4, 0), B 1 (0, –2), B 2 (0, 2). Since is half the interfocal distance, the points are the foci of the ellipse. Let's calculate the eccentricity:

Headmistresses D 1 , D 2 are described by the equations:

Draw an ellipse (Fig. 9.4).

Example 2. Define ellipse parameters

Solution. Let's compare this equation with the canonical equation of an ellipse with a displaced center. Finding the center of the ellipse WITH: Semi-major axis, semi-minor axis, straight lines – major axes. Half the interfocal distance and therefore the focal points Eccentricity of the Directrix D 1 and D 2 can be described using the equations: (Fig. 9.5).

Example 3. Determine which curve is given by the equation and draw it:

1) x 2 + y 2 + 4x – 2y + 4 = 0; 2) x 2 + y 2 + 4x – 2y + 6 = 0;

3) x 2 + 4y 2 – 2x + 16y + 1 = 0; 4) x 2 + 4y 2 – 2x + 16y + 17 = 0;

Solution. 1) Let us reduce the equation to canonical form by isolating the complete square of the binomial:

x 2 + y 2 + 4x – 2y + 4 = 0;

(x 2 + 4x) + (y 2 – 2y) + 4 = 0;

(x 2 + 4x + 4) – 4 + (y 2 – 2y + 1) – 1 + 4 = 0;

(x + 2) 2 + (y – 1) 2 = 1.

Thus, the equation can be reduced to the form

(x + 2) 2 + (y – 1) 2 = 1.

This is the equation of a circle with center at point (–2, 1) and radius R= 1 (Fig. 9.6).

2) We select the perfect squares of the binomials on the left side of the equation and get:

(x + 2) 2 + (y – 1) 2 = –1.

This equation does not make sense on the set of real numbers, since the left side is non-negative for any real values ​​of the variables x And y, and the right one is negative. Therefore, they say that this is the equation of an “imaginary circle” or that it defines an empty set of points in the plane.

3) Select complete squares:

x 2 + 4y 2 – 2x + 16y + 1 = 0;

(x 2 – 2x + 1) – 1 + 4(y 2 + 4y + 4) – 16 + 1 = 0;

(x – 1) 2 + 4(y + 2) 2 – 16 = 0;

(x – 1) 2 + 4(y + 2) 2 = 16.

So the equation looks like:

The resulting equation, and therefore the original one, defines an ellipse. The center of the ellipse is at the point ABOUT 1 (1, –2), the main axes are given by the equations y = –2, x= 1, and the semimajor axis A= 4, minor axis b= 2 (Fig. 9.7).

4) After selecting complete squares we have:

(x – 1) 2 + 4(y+ 2) 2 – 17 + 17 = 0 or ( x – 1) 2 + 4(y + 2) 2 = 0.

The resulting equation specifies a single point on the plane with coordinates (1, –2).

5) Let's bring the equation to canonical form:

Obviously, it defines an ellipse, the center of which is located at the point the main axes are given by the equations with the semi-major axis and the semi-minor axis (Fig. 9.8).

Example 4. Write down the equation of the tangent to a circle of radius 2 centered at the right focus of the ellipse x 2 + 4y 2 = 4 at the point of intersection with the y-axis.

Solution. Let us reduce the ellipse equation to canonical form (9.7):

This means that the right focus is also - Therefore, the required equation for a circle of radius 2 has the form (Fig. 9.9):

The circle intersects the ordinate axis at points whose coordinates are determined from the system of equations:

We get:

Let these be points N(0; –1) and M(0; 1). This means that we can construct two tangents, let’s denote them T 1 and T 2. According to the well-known property, a tangent is perpendicular to the radius drawn to the point of contact.

Let Then the tangent equation T 1 will take the form:

So, either T 1: It is equivalent to the equation