An example of constructing an ellipse given by an equation. Ellipse property definition construction

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.

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.

Studying the shape of an ellipse using its equation

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

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

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°.

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 a line given by a general equation is , ellipse.

Solution. We transform the general equation. We use the transfer of the free term to the right side, the term-by-term division of the equation by the same number and the reduction of fractions:

Answer. The equation obtained as a result of the transformations is the canonical equation of the ellipse. Therefore, this line is an ellipse.

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

Solution. We look at the formula for the canonical equation of an ellipse and substitute: the semimajor axis is a= 5, the semiminor axis is 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 symmetrically about the origin. Compose the canonical equation of the ellipse if:

1) the distance between the focuses 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.

Lectures on algebra and geometry. Semester 1.

Lecture 15. Ellipse.

Chapter 15. Ellipse.

clause 1. Basic definitions.

Definition. An ellipse is the GMT of a plane, the sum of the distances to two fixed points of the plane, called foci, is a constant value.

Definition. The distance from an arbitrary point M of the plane to the focus of the ellipse is called the focal radius of the point M.

Designations:
– foci of the ellipse,
– focal radii of point M.

By the definition of an ellipse, a point M is a point of an ellipse if and only if
– constant value. This constant is usually denoted as 2a:

. (1)

notice, that
.

By definition of an ellipse, its foci are fixed points, so the distance between them is also a constant value for a given ellipse.

Definition. The distance between the foci of the ellipse is called the focal length.

Designation:
.

From a triangle
follows that
, i.e.

.

Let us denote by b the number equal to
, i.e.

. (2)

Definition. Attitude

(3)

is called the eccentricity of the ellipse.

Let us introduce a coordinate system on this plane, which we will call canonical for the ellipse.

Definition. The axis on which the foci of the ellipse lie is called the focal axis.

Let's construct a canonical PDSC for the ellipse, see Fig. 2.

We select the focal axis as the abscissa axis, and draw the ordinate axis through the middle of the segment
perpendicular to the focal axis.

Then the foci have coordinates
,
.

clause 2. Canonical equation of an ellipse.

Theorem. In the canonical coordinate system for an ellipse, the equation of the ellipse has the form:

. (4)

Proof. We carry out the proof in two stages. At the first stage, we will prove that the coordinates of any point lying on the ellipse satisfy equation (4). At the second stage we will prove that any solution to equation (4) gives the coordinates of a point lying on the ellipse. From here it will follow that equation (4) is satisfied by those and only those points of the coordinate plane that lie on the ellipse. From this and from the definition of the equation of a curve it will follow that equation (4) is an equation of an ellipse.

1) Let the point M(x, y) be a point of the ellipse, i.e. the sum of its focal radii is 2a:

.

Let's use the formula for the distance between two points on the coordinate plane and use this formula to find the focal radii of a given point M:

,
, from where we get:

Let's move one root to the right side of the equality and square it:

Reducing, we get:

We present similar ones, reduce by 4 and remove the radical:

.

Squaring

Open the brackets and shorten by
:

where we get:

Using equality (2), we obtain:

.

Dividing the last equality by
, we obtain equality (4), etc.

2) Let now a pair of numbers (x, y) satisfy equation (4) and let M(x, y) be the corresponding point on the coordinate plane Oxy.

Then from (4) it follows:

.

We substitute this equality into the expression for the focal radii of point M:

.

Here we used equality (2) and (3).

Thus,
. Likewise,
.

Now note that from equality (4) it follows that

or
etc.
, then the inequality follows:

.

From here it follows, in turn, that

or
And

,
. (5)

From equalities (5) it follows that
, i.e. the point M(x, y) is a point of the ellipse, etc.

The theorem has been proven.

Definition. Equation (4) is called the canonical equation of the ellipse.

Definition. The canonical coordinate axes for an ellipse are called the principal axes of the ellipse.

Definition. The origin of the canonical coordinate system for an ellipse is called the center of the ellipse.

clause 3. Properties of the ellipse.

Theorem. (Properties of an ellipse.)

1. In the canonical coordinate system for an ellipse, everything

the points of the ellipse are in the rectangle

,
.

2. The points lie on

3. An ellipse is a curve that is symmetrical with respect to

their main axes.

4. The center of the ellipse is its center of symmetry.

Proof. 1, 2) Immediately follows from the canonical equation of the ellipse.

3, 4) Let M(x, y) be an arbitrary point of the ellipse. Then its coordinates satisfy equation (4). But then the coordinates of the points also satisfy equation (4), and, therefore, are points of the ellipse, from which the statements of the theorem follow.

The theorem has been proven.

Definition. The quantity 2a is called the major axis of the ellipse, the quantity a is called the semi-major axis of the ellipse.

Definition. The quantity 2b is called the minor axis of the ellipse, the quantity b is called the semiminor axis of the ellipse.

Definition. The points of intersection of an ellipse with its main axes are called the vertices of the ellipse.

Comment. An ellipse can be constructed as follows. On the plane, we “hammer a nail into the focal points” and fasten a thread length to them
. Then we take a pencil and use it to stretch the thread. Then we move the pencil lead along the plane, making sure that the thread is taut.

From the definition of eccentricity it follows that

Let us fix the number a and direct the number c to zero. Then at
,
And
. In the limit we get

or
– equation of a circle.

Let us now direct
. Then
,
and we see that in the limit the ellipse degenerates into a straight line segment
in the notation of Figure 3.

clause 4. Parametric equations of the ellipse.

Theorem. Let
– arbitrary real numbers. Then the system of equations

,
(6)

are parametric equations of an ellipse in the canonical coordinate system for the ellipse.

Proof. It is enough to prove that the system of equations (6) is equivalent to equation (4), i.e. they have the same set of solutions.

1) Let (x, y) be an arbitrary solution to system (6). Divide the first equation by a, the second by b, square both equations and add:

.

Those. any solution (x, y) of system (6) satisfies equation (4).

2) Conversely, let the pair (x, y) be a solution to equation (4), i.e.

.

From this equality it follows that the point with coordinates
lies on a circle of unit radius with center at the origin, i.e. is a point on a trigonometric circle to which a certain angle corresponds
:

From the definition of sine and cosine it immediately follows that

,
, Where
, from which it follows that the pair (x, y) is a solution to system (6), etc.

The theorem has been proven.

Comment. An ellipse can be obtained as a result of uniform “compression” of a circle of radius a towards the abscissa axis.

Let
– equation of a circle with center at the origin. “Compression” of a circle to the abscissa axis is nothing more than a transformation of the coordinate plane, carried out according to the following rule. For each point M(x, y) we associate a point on the same plane
, Where
,
– compression ratio.

With this transformation, each point on the circle “transitions” to another point on the plane, which has the same abscissa, but a smaller ordinate. Let's express the old ordinate of a point through the new one:

and substitute circles into the equation:

.

From here we get:

. (7)

It follows from this that if before the “compression” transformation the point M(x, y) lay on the circle, i.e. its coordinates satisfied the equation of the circle, then after the “compression” transformation this point “transformed” into the point
, whose coordinates satisfy the ellipse equation (7). If we want to obtain the equation of an ellipse with semiminor axisb, then we need to take the compression factor

.

clause 5. Tangent to an ellipse.

Theorem. Let
– arbitrary point of the ellipse

.

Then the equation of the tangent to this ellipse at the point
has the form:

. (8)

Proof. It is enough to consider the case when the point of tangency lies in the first or second quarter of the coordinate plane:
. The equation of the ellipse in the upper half-plane has the form:

. (9)

Let's use the tangent equation to the graph of the function
at the point
:

Where
– the value of the derivative of a given function at a point
. The ellipse in the first quarter can be considered as a graph of function (8). Let's find its derivative and its value at the point of tangency:

,

. Here we took advantage of the fact that the tangent point
is a point of the ellipse and therefore its coordinates satisfy the ellipse equation (9), i.e.

.

We substitute the found value of the derivative into the tangent equation (10):

,

where we get:

This implies:

Let's divide this equality by
:

.

It remains to note that
, because dot
belongs to the ellipse and its coordinates satisfy its equation.

The tangent equation (8) is proved in a similar way at the point of tangency lying in the third or fourth quarter of the coordinate plane.

And finally, we can easily verify that equation (8) gives the tangent equation at the points
,
:

or
, And
or
.

The theorem has been proven.

clause 6. Mirror property of an ellipse.

Theorem. The tangent to the ellipse has equal angles with the focal radii of the point of tangency.

Let
– point of contact,
,
– focal radii of the tangent point, P and Q – projections of foci on the tangent drawn to the ellipse at the point
.

The theorem states that

. (11)

This equality can be interpreted as the equality of the angles of incidence and reflection of a ray of light from an ellipse released from its focus. This property is called the mirror property of the ellipse:

A ray of light released from the focus of the ellipse, after reflection from the mirror of the ellipse, passes through another focus of the ellipse.

Proof of the theorem. To prove the equality of angles (11), we prove the similarity of triangles
And
, in which the parties
And
will be similar. Since the triangles are right-angled, it is enough to prove the equality