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

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

Parametric equation of an ellipse

Parametric equation of an ellipse in the canonical coordinate system has the form

\begin(cases)x=a\cdot\cos(t),\\ y=b\cdot\sin(t),\end(cases)0\leqslant t<2\pi.

Indeed, substituting these expressions into equation (3.49), we arrive at the basic trigonometric identity \cos^2t+\sin^2t=1 .

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

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

\frac(1^2)(2^2)+\frac(y^2)(1^2)=1 \quad \Leftrightarrow \quad y^2=\frac(3)(4) \quad \Leftrightarrow \ quad y=\pm\frac(\sqrt(3))(2).

Therefore, points with coordinates \left(1;\,\frac(\sqrt(3))(2)\right)\!,~\left(1;\,-\frac(\sqrt(3))(2)\right)- belong to an ellipse.

Calculate the compression ratio k=\frac(b)(a)=\frac(1)(2); focal length 2c=2\sqrt(a^2-b^2)=2\sqrt(2^2-1^2)=2\sqrt(3); eccentricity e=\frac(c)(a)=\frac(\sqrt(3))(2); focal parameter p=\frac(b^2)(a)=\frac(1^2)(2)=\frac(1)(2). We compose the directrix equations: x=\pm\frac(a^2)(c)~\Leftrightarrow~x=\pm\frac(4)(\sqrt(3)).

Lectures on Algebra and Geometry. Semester 1.

Lecture 15. Ellipse.

Chapter 15

item 1. Basic definitions.

Definition. An ellipse is the GMT of a plane, the sum of the distances of which 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:
are the foci of the ellipse,
are the focal radii of the point M.

By definition of an ellipse, a point M is a point of the ellipse if and only if
is a 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 the given ellipse.

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

Designation:
.

From a triangle
follows that
, i.e.

.

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 the given 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 the canonical PDSC for the ellipse, see Fig.2.

We choose 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
,
.

item 2. Canonical equation of an ellipse.

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

. (4)

Proof. We will 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 of 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 here and from the definition of the curve equation, it will follow that equation (4) is an ellipse equation.

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

.

We use the formula for the distance between two points on the coordinate plane and find the focal radii of a given point M using this formula:

,
, from where we get:

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

Reducing, we get:

We give similar ones, reduce by 4 and isolate the radical:

.

We square

Open the brackets and shorten
:

from where we get:

Using equality (2), we obtain:

.

Dividing the last equality by
, we obtain equality (4), p.t.d.

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

Then from (4) it follows:

.

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

.

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

In this way,
. Likewise,
.

Now note that it follows from equality (4) that

or
and because
, then the following inequality follows:

.

From this, in turn, it follows that

or
and

,
. (5)

It follows from equalities (5) 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 the 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.

item 3. Ellipse properties.

Theorem. (Properties of an ellipse.)

1. In the canonical coordinate system for the ellipse, all

the points of the ellipse are in the rectangle

,
.

2. Points lie on

3. An ellipse is a curve symmetrical about

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 the 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 major semiaxis of the ellipse.

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

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

Comment. An ellipse can be constructed in the following way. On a plane, we “hammer a nail” into the tricks and fasten a thread of 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 in a taut state.

From the definition of eccentricity it follows that

We fix a number a and let c tend to zero. Then at
,
and
. In the limit we get

or
is the circle equation.

Let's strive now
. Then
,
and we see that in the limit the ellipse degenerates into a line segment
in the notation of Figure 3.

item 4. Parametric equations of an ellipse.

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

,
(6)

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

Proof. It suffices 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 of 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.

.

It follows from this equality that the point with coordinates
lies on a circle of unit radius centered at the origin, i.e. is a point of the trigonometric circle, which corresponds to some angle
:

From the definition of sine and cosine, it immediately follows that

,
, where
, whence 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 a uniform "compression" of a circle of radius a to the abscissa axis.

Let
is the equation of a circle centered at the origin. The "compression" of the circle to the abscissa axis is nothing more than the transformation of the coordinate plane, carried out according to the following rule. To each point M(x, y) we put in correspondence a point of the same plane
, where
,
is the "compression" factor.

With this transformation, each point of the circle "passes" to another point in the plane, which has the same abscissa, but a smaller ordinate. Let's express the old ordinate of the point in terms of the new one:

and substitute into the circle 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 circle equation, then after the "compression" transformation, this point "passed" into the point
, whose coordinates satisfy the ellipse equation (7). If we want to get the equation of an ellipse with a minor semi-axis b, then we need to take the compression factor

.

item 5. Tangent to an ellipse.

Theorem. Let
- arbitrary point of the ellipse

.

Then the equation of the tangent to this ellipse at the point
looks like:

. (8)

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

. (9)

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

where
is the value of the derivative of this function at the point
. The ellipse in the first quarter can be viewed as a graph of function (8). Let's find its derivative and its value at the point of contact:

,

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

.

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

,

from where we get:

This implies:

Let's divide this equation into
:

.

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

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

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

or
, and
or
.

The theorem has been proven.

item 6. The mirror property of an ellipse.

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

Let
- point of contact
,
are the focal radii of the tangent point, P and Q are the projections of the 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 light beam from an ellipse released from its focus. This property is called the mirror property of the ellipse:

A beam of light emitted 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 sides
and
will be similar. Since the triangles are right-angled, it suffices to prove the equality

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

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

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

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

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

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

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

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

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

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

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

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

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

After opening the brackets and reducing like terms, we get

√((x + c) 2 + y 2) = a + εx

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

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

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

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

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

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

which after transformations leads to the equation

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

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

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

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

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

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

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

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

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

and each of these equations is an ellipse equation.

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

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

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

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

|F 1 M|/(а/ε - x) = ε

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

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

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

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

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

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

Curves of the second order on a plane are called lines defined by equations in which the variable coordinates x and y 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 is not equal to zero.

When solving problems with second-order curves, the canonical equations of an ellipse, hyperbola, and parabola are most often considered. It is easy to pass 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 in the plane, those for which the sum of the distances to the points, called foci, is a constant and greater than the distance between the foci.

Focuses are marked as in the figure below.

The canonical equation of an ellipse is:

where a and b (a > b) - the lengths of the semiaxes, 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 the ellipse is a straight line passing through the middle of the segment perpendicular to this segment. Dot O 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 points ( a, O) and (- a, O), and the y-axis is at points ( b, O) and (- b, O). These four points are called the vertices of the ellipse. The segment between the vertices of the ellipse on the abscissa axis is called its major axis, and on the ordinate axis - the minor axis. Their segments from the top to the center of the ellipse are called semiaxes.

If a a = b, then the equation of the ellipse takes the form . This is the equation for a circle of radius a, and the circle special case 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 by the general equation , an ellipse.

Solution. We make transformations of the general equation. We apply 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 the canonical equation of the ellipse. Therefore, this line is an ellipse.

Example 2 Write the canonical equation of an ellipse if its semiaxes are 5 and 4, respectively.

Solution. We look at the formula for the canonical equation of the ellipse and substitute: the semi-major axis is a= 5 , the minor semiaxis is b= 4 . We get the canonical equation of the ellipse:

Points and marked in green on the major axis, where

called tricks.

called eccentricity ellipse.

Attitude b/a characterizes the "oblateness" of the ellipse. The smaller this ratio, the more the ellipse is extended along the major axis. However, the degree of elongation of the ellipse is more often expressed in terms of eccentricity, the formula of which is given above. For different ellipses, the eccentricity varies from 0 to 1, always remaining less than one.

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

Solution. We make simple conclusions:

If the major axis is 10, then its half, i.e. semiaxis a = 5 ,

If the distance between foci is 8, then the number c of the focus coordinates is 4.

Substitute and calculate:

The result is the canonical equation of the ellipse:

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

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

.

We calculate the square of the length of the minor semiaxis:

We compose the canonical equation of the ellipse:

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

Solution. Need to find a number c, which defines the first coordinates of the foci of the ellipse:

.

We get the focuses of the ellipse:

Example 6 The foci of the ellipse are located on the axis Ox symmetrical about the origin. Write the canonical equation of an ellipse if:

1) the distance between the foci is 30, and the major axis is 34

2) the minor axis is 24, and one of the focuses is at the point (-5; 0)

3) eccentricity, and one of the foci is at the point (6; 0)

We continue to solve problems on the ellipse together

If - an arbitrary point of the ellipse (marked in green in the drawing in the upper right part of the ellipse) and - the distances 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.

Straight lines defined by equations

called directors ellipse (in the drawing - red lines along the edges).

From the above two equations 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 into the directrix equation and find that it is required to find the eccentricity of the ellipse, i.e. . All data for this is. We calculate:

.

We get the equation of the directrix of the ellipse:

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