Find the length of the major axis of the ellipsoid. Lines of the second order. Ellipse and its canonical equation. Circle
11.1. Basic concepts
Consider the lines defined by equations of the second degree with respect to the current coordinates
The coefficients of the equation are real numbers, but at least one of the numbers A, B, or C is non-zero. Such lines are called lines (curves) of the second order. It will be established below that equation (11.1) defines a circle, ellipse, hyperbola, or parabola in the plane. Before proceeding to this assertion, let us study the properties of the enumerated curves.
11.2. Circle
The simplest curve of the second order is a circle. Recall that a circle of radius R centered at a point is the set of all points Μ of the plane that satisfy the condition . Let a point in a rectangular coordinate system have coordinates x 0, y 0 a - an arbitrary point of 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 the given circle and is not satisfied by the coordinates of any point that does not lie on the circle.
Equation (11.2) is called the canonical equation of the circle
In particular, assuming and , we obtain the equation of a circle centered 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 see that two conditions are satisfied for the equation of a circle:
1) the coefficients at x 2 and y 2 are equal to each other;
2) there is no member containing the xy product of the current coordinates.
Let's consider the inverse problem. Putting in equation (11.1) the values and , 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 a 
, then equation (11.4), and therefore the equivalent equation (11.3), will not determine any line, since the right side of equation (11.4) is negative, and the left side is not negative (say: “imaginary circle”).
11.3. Ellipse
Canonical equation of an ellipse
Ellipse is the set of all points of the plane, the sum of the distances from each of them to two given points of this plane, called tricks , is a constant value greater than the distance between the foci.
Denote the foci by F1 and F2, the distance between them in 2 c, and the sum of the distances from an arbitrary point of the ellipse to the foci - through 2 a(see fig. 49). By definition 2 a > 2c, i.e. a
> c.
To derive the equation of an ellipse, we choose a coordinate system so that the foci F1 and F2 lie on the axis , and the origin coincides with the midpoint 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 fact, is the equation of an ellipse.
We transform equation (11.5) to a simpler form as follows:
Because a>With, then . Let's put
(11.6)
Then the last equation takes the form or
(11.7)
It can be proved that equation (11.7) is equivalent to the original equation. It's called the canonical equation of the ellipse .
Ellipse is a curve of the second order.
Study of the shape of an ellipse according to its equation
Let's 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 points ,, also belong to it. It follows that the ellipse is symmetrical with respect to the axes and , 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 ,
A2 , B1, B2 called the vertices of the ellipse. Segments A 1 A2 and B1 B2, as well as their lengths 2 a and 2 b are called respectively major and minor axes ellipse. Numbers a and b are called large and small, respectively. axle shafts ellipse.
3. It follows from equation (11.7) that each term on the left-hand side does not exceed one, i.e. there are inequalities and or and . Therefore, 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, that is, if it increases, then it decreases and vice versa.
From what has been said, it follows that the ellipse has the shape shown in Fig. 50 (oval closed curve).
More about the ellipse
The shape of the ellipse depends on the ratio. When the ellipse turns into a circle, the ellipse equation (11.7) takes the form . As a characteristic of the shape of an ellipse, the ratio is more often used. 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 oblate the ellipse will be; if we put 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,
There are formulas
Straight 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:
It follows from equality (11.6) that . If , then equation (11.7) defines an ellipse, the major axis of which lies on the Oy axis, and the minor axis lies on the Ox axis (see Fig. 52). The foci of such an ellipse are at the points and , where 
.
11.4. Hyperbola
Canonical equation of a hyperbola
Hyperbole the set of all points of the plane is called, the modulus of the difference in distances from each of which to two given points of this plane, called tricks , is a constant value, smaller than the distance between the foci.
Denote the foci by F1 and F2 the distance between them through 2s, and the modulus of the difference in distances from each point of the hyperbola to the foci through 2a. By definition 2a < 2s, i.e. a < c.
To derive the hyperbola equation, we choose a coordinate system so that the foci F1 and F2 lie on the axis , and the origin coincided with the midpoint 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 ellipse equation, we get
canonical equation of a hyperbola
(11.9)
(11.10)
A hyperbola is a line of the second order.
Investigation of the form of a hyperbola according to its equation
Let us establish the shape of the hyperbola using its caconic equation.
1. Equation (11.9) contains x and y only in even powers. Therefore, the hyperbola is symmetrical with respect to the axes and , as well as with respect to the point , which is called the center of the hyperbola.
2. Find the intersection points 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 obtain , which cannot be. Therefore, the hyperbola does not intersect the y-axis.
The points and are called peaks hyperbolas, and the segment
real axis , line segment - real semiaxis hyperbole.
The line segment connecting the points is called imaginary axis , number b - imaginary axis . Rectangle with sides 2a and 2b called the main rectangle of a hyperbola .
3. It follows from equation (11.9) 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 (the right branch of the hyperbola) and to the left of the line (the left branch of the hyperbola).
4. From the equation (11.9) of the hyperbola, it can be seen that when it increases, then it also increases. This follows from the fact that the difference keeps a constant value equal to one.
It follows from what has been said that the hyperbola has the shape shown in Figure 54 (a curve consisting of two unbounded branches).
Asymptotes of a hyperbola
The line L is called the asymptote 
of an unbounded curve K if the distance d from point M of curve K to this line tends to zero as the point M moves along the curve K indefinitely from the origin. Figure 55 illustrates the concept of an asymptote: the line L is an asymptote for the curve K.
Let us show that the hyperbola has two asymptotes:
(11.11)
Since the lines (11.11) and the hyperbola (11.9) are symmetrical with respect to the coordinate axes, it suffices to consider only those points of the indicated lines that are located in the first quadrant.
Take on a straight line a point N having the same abscissa x as a point on a hyperbola 
(see Fig. 56), and find the difference ΜN 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; numerator is a constant value. Therefore, the length of the segment 
ΜN tends to zero. Since ΜN is greater than the distance d from the point Μ to the line, then d even more so tends to zero. Thus, 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 lines passing through the opposite vertices of this rectangle - the asymptotes of the hyperbola and mark the vertices and , hyperbola.
The equation of an equilateral hyperbola.
whose asymptotes are the coordinate axes
Hyperbola (11.9) is called equilateral if its semiaxes are equal (). Its canonical equation
(11.12)
The asymptotes of an equilateral hyperbola have equations and are therefore bisectors of the coordinate angles.
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 the rotation of the coordinate axes:
We substitute the values of x and y in equation (11.12):
The equation of an equilateral hyperbola, for which the axes Ox and Oy are asymptotes, will have the form .
More 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, it follows from equality (11.10) that i.e. and 
.
This shows that the smaller the eccentricity of the hyperbola, the smaller the ratio - of its semi-axes, which means that the more its main rectangle is extended.
The eccentricity of an equilateral hyperbola is . Really,
Focal radii 
and 
for the points of the right branch of the hyperbola have the form and , and for the left - 
and 
.
Straight lines are called directrixes of a hyperbola. Since for the hyperbola ε > 1, then . This means that the right directrix is located between the center and the right vertex of the hyperbola, the left directrix is 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.
Obviously, the hyperbolas and have common asymptotes. Such hyperbolas are called conjugate.
11.5. Parabola
Canonical parabola equation
A parabola is the set of all points in a 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 parabola equation, we choose the Oxy coordinate system so that the Oxy axis passes through the focus F perpendicular to the directrix in the direction from the directrix to F, and the origin O is located in the middle between the focus and the directrix (see Fig. 60). In the selected system, the focus F has coordinates , and the directrix equation has the form , or .
1. In equation (11.13), the variable y is included in an even degree, which means that the parabola is symmetric about the Ox axis; the x-axis is the axis of symmetry of the parabola.
2. Since ρ > 0, it follows from (11.13) that . Therefore, the parabola is located to the right of the y-axis.
3. When we have y \u003d 0. Therefore, the parabola passes through the origin.
4. With an unlimited increase in x, the module y also increases indefinitely. The parabola has the form (shape) shown in Figure 61. The point O (0; 0) is called the vertex of the parabola, the segment FM \u003d r is called the focal radius of the point M.
Equations , , ( p>0) also define parabolas, they are shown in Figure 62
It is easy to show that the graph of a square trinomial, where , B and C are any real numbers, is a parabola in the sense of its definition above.
11.6. General equation of second order lines
Equations of curves of the second order with axes of symmetry parallel to the coordinate axes
Let us first find the equation of an ellipse centered at a point whose symmetry axes are parallel to the coordinate axes Ox and Oy and the semiaxes are respectively equal to a and b. Let us place in the center of the ellipse O 1 the origin of the new coordinate system , whose axes and semi-axes a and b(see fig. 64):
And 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 in one direction, bring like terms, introduce new notation for the coefficients) can be written using a single equation of the form
where the coefficients A and C are not equal to zero at the same time.
The question arises: does any 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 equation of the second order
Consider now the 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 in it.
Using formulas for turning axes
Let's express the old coordinates in terms of the new ones:
We choose the angle a so that the coefficient at x "y" vanishes, i.e., so that the equality
Thus, when the axes are rotated through an angle a that satisfies condition (11.17), equation (11.15) reduces to equation (11.14).
Conclusion: the general equation of the second order (11.15) defines on the plane (except for the cases of degeneration and decay) the following curves: circle, ellipse, hyperbola, parabola.
Note: If A = C, then equation (11.17) loses its meaning. In this case cos2α = 0 (see (11.16)), then 2α = 90°, i.e. α = 45°. So, at A = C, the coordinate system should be rotated by 45 °.
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 indicated family. In this case, no two ellipses intersect, since the sum of the focal radii uniquely determines a specific ellipse. So, the described family of ellipses without intersections covers the entire plane, except for the points of the segment F 1 F 2 . 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 common solutions. However, it is easy to verify that the system
for ã ≠ a has no solutions. To do this, it is enough to exclude, for example, x from the first equation:
which after transformations leads to the equation
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 discussed above gives a sufficient idea of the appearance of an 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.
Lines of the second order.
Ellipse and its canonical equation. Circle
After a thorough study straight lines on the plane we continue to study the geometry of the two-dimensional world. The stakes are doubled and I invite you to visit the picturesque gallery of ellipses, hyperbolas, parabolas, which are typical representatives of second order lines. The tour 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 ( is a real number, are 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" and "y" in integer non-negative degrees.
Line order is 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 the ease of being, we consider that all subsequent calculations take place in Cartesian coordinates.
General Equation the second-order line has the form , where 
are arbitrary real numbers (it is customary to write with a multiplier - "two"), and the coefficients are not simultaneously equal to zero.
If , then the equation simplifies to 
, and if the coefficients are not simultaneously equal to zero, then this is exactly general equation of a "flat" straight line, which represents first order line.
Many understood the meaning of the new terms, but, nevertheless, in order to 100% assimilate the material, we stick our fingers into the socket. To determine the line order, iterate over all terms its equations and for each of them find sum of powers incoming variables.
For example:
the term contains "x" to the 1st degree;
the term contains "Y" to the 1st degree;
there are no variables in the term, so the sum of their powers is zero.
Now let's figure out why the equation sets the line second order:
the term contains "x" in the 2nd degree;
the term has the sum of the degrees of the variables: 1 + 1 = 2;
the term contains "y" in the 2nd degree;
all other terms - lesser degree.
Maximum value: 2
If we additionally add to our equation, say, , then it will already determine third order line. It is obvious that the general form of the 3rd order line equation contains a “complete set” of terms, the sum of the degrees of variables in which is equal to three:
, where the coefficients are not simultaneously equal to zero.
In the event that one or more suitable terms are added that contain 
, then we will talk about 4th order lines, etc.
We will have to deal with algebraic lines of the 3rd, 4th and higher orders more than once, in particular, when getting acquainted with polar coordinate system.
However, let us return to the general equation and recall its simplest school variations. Examples are the parabola, whose equation can be easily reduced to a general form, and the 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 will not immediately realize that this is hyperbole. Such layouts are good only at a masquerade, therefore, in the course of analytical geometry, a typical problem is considered reduction of the 2nd order line equation to the canonical form.
What is the canonical form of an equation?
This is the generally accepted standard form of the 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 simply visible.
Obviously, any 1st order line represents a straight line. On the second floor, there is no longer a janitor waiting for us, but a much more diverse company of nine statues:
Classification of second order lines
With the help of a special set of actions, any second-order line equation is reduced to one of the following types:
( and are positive real numbers)
1) 
is the canonical equation of the ellipse;
2) is the canonical equation of the hyperbola;
3) 
is the canonical equation of the parabola;
4) – imaginary ellipse;
5) - a pair of intersecting lines;
6) - couple imaginary intersecting lines (with the only real point of intersection at the origin);
7) - a pair of parallel lines;
8) - couple imaginary parallel lines;
9) is a pair of coinciding lines.
Some readers may get the impression that the list is incomplete. For example, in paragraph number 7, the equation sets the pair direct, parallel to the axis, and the question arises: where is the equation that determines the lines parallel to the y-axis? Answer: it not considered canon. The straight lines represent the same standard case rotated by 90 degrees, and the additional entry in the classification is redundant, since it does not carry 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 "elebs eccentricity".
The canonical equation of an ellipse has the form , where are positive real numbers, and . I will formulate the definition of an ellipse later, but for now it's time to take a break from talking and solve a common problem:
How to build an ellipse?
Yes, take it and just draw it. The assignment is common, and a significant part of the students do not quite competently cope with the drawing:
Example 1
Construct an ellipse given by the equation
Solution: first we bring the equation to the canonical form: 
Why bring? One of the advantages of the canonical equation is that it allows you to instantly determine ellipse vertices, which are at the 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 axis ellipse;
number 
– semi-minor axis.
in our example: .
To quickly imagine what this or that ellipse looks like, just look at the values \u200b\u200bof "a" and "be" of its canonical equation.
Everything is fine, neat and beautiful, but there is one caveat: I completed the drawing using the program. And you can draw with any application. However, in harsh reality, a checkered piece of paper lies on the table, and mice dance around our hands. People with artistic talent, of course, can argue, but you also have mice (albeit smaller ones). It is not in vain that mankind invented a ruler, a compass, a 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. Still all right, if the ellipse is small, for example, with semiaxes. Alternatively, you can reduce the scale and, accordingly, the dimensions of the drawing. But in the general case it is highly desirable to find additional points.
There are two approaches to constructing an ellipse - geometric and algebraic. I don’t like building with a compass and ruler because of the short algorithm and the significant clutter of the drawing. 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 ellipse equation on the draft, we quickly express: 
The equation is then split into two functions: 
– defines the upper arc of the ellipse; 
– defines the lower arc of the ellipse.
The ellipse given by the canonical equation is symmetrical with respect to the coordinate axes, as well as with respect to the origin. And that's great - symmetry is almost always a harbinger of a freebie. Obviously, it is enough to deal with the 1st coordinate quarter, so we need a function 
. It suggests finding additional points with abscissas 
. We hit three SMS on the calculator: 
Of course, it is also pleasant that if a serious error is made in the calculations, then this will immediately become clear during the construction.
Mark points on the drawing (red color), symmetrical points on the other arcs (blue color) and carefully connect the whole company with a line: 
It is better to draw the initial sketch thinly and thinly, and only then apply pressure to the pencil. The result should be quite a 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 with a detailed formulation. The purpose of this lesson is not to consider the theory of ovals and their various types, which are practically not given attention in the standard course of analytic geometry. And, in accordance with more current needs, we immediately go to the strict definition of an ellipse:
Ellipse- this 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 value, numerically equal to the length of the major axis of this ellipse: .
In this case, the distance between the foci is less than this value: .
Now it will become clearer: 
Imagine that the blue dot "rides" on 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 "em" in the right vertex of the ellipse, then: , which was required to be checked.
Another way to draw an ellipse is based on the definition of an ellipse. Higher mathematics, at times, is the cause of tension and stress, so it's time to have another session of unloading. Please take a piece of 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 neck of the pencil will be at some point, which belongs to the ellipse. Now begin to guide the pencil across the sheet of paper, keeping the green thread very taut. Continue the process until you return to the starting point ... excellent ... the drawing can be submitted for verification by the doctor to the teacher =)
How to find the focus of an ellipse?
In the above example, I depicted "ready" focus points, and now we will learn how to extract them from the depths of geometry.
If the ellipse is given by the canonical equation , then its foci have coordinates 
, where is it distance from each of the foci to the center of symmetry of the ellipse.
Calculations are easier than steamed turnips: 
! With the meaning "ce" it is impossible to identify the specific coordinates of tricks! I repeat, 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 cannot be tied to the canonical position of the ellipse either. 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 bear this in mind as you explore the topic further.
The eccentricity of an ellipse and its geometric meaning
The eccentricity of an ellipse is a ratio that can take values within .
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 semi-major axis will remain constant. Then the eccentricity formula will take the form: .
Let's start to approximate the value of the eccentricity to unity. This is only possible if . What does it mean? ...remembering tricks 
. This means that the foci of the ellipse will "disperse" 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.
In this way, the closer the eccentricity of the ellipse is to one, the more oblong the ellipse is.
Now let's simulate the opposite process: the foci of the ellipse 
went towards each other, approaching the center. This means that the value of "ce" is getting smaller and, accordingly, the eccentricity tends to zero: .
In this case, the “green segments”, on the contrary, will “become crowded” and they will begin to “push” the line of the ellipse up and down.
In this way, the closer the eccentricity value is to zero, the more the ellipse looks like... 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 semiaxes, the canonical equation of the ellipse takes the form, which reflexively transforms to the well-known circle equation from the school with the center at the origin of the radius "a".
In practice, the notation with the “speaking” letter “er” is more often used:. The radius is called the length of the segment, while each point of the circle is removed from the center by the distance of the radius.
Note that the definition of an ellipse remains completely correct: the foci matched, and the sum of the lengths of the matched segments for each point on the circle is a constant value. Since the distance between foci is the eccentricity of any circle is zero.
A circle is built easily and quickly, it is enough to arm yourself with 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 a cheerful Matan's form:
is the function of the upper semicircle;
is the function of the lower semicircle.
Then we find the desired values, differentiable, integrate and do other good things.
The article, of course, is for reference only, but how can one live without love in the world? Creative task for independent solution
Example 2
Compose the canonical equation of an ellipse if one of its foci and the semi-minor axis are known (the center is at the origin). Find vertices, additional points and draw a line on the drawing. Calculate the eccentricity.
Solution and drawing at the end of the lesson
Let's add an action:
Rotate and translate an ellipse
Let's return to the canonical equation of the ellipse, namely, to the condition, the riddle of which has been tormenting inquisitive minds since the first mention of this curve. Here we have considered an ellipse 
, but in practice can not the equation 
? After all, here, however, it seems to be like an ellipse too!
Such an equation is rare, but it does come across. And it does define an ellipse. Let's dispel the mystic: 
As a result of the construction, our native ellipse is obtained, rotated by 90 degrees. That is, 
- this is non-canonical entry ellipse 
. Record!- the equation 
does not specify any other ellipse, since there are no points (foci) on the axis that would satisfy the definition of an ellipse.
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. An ellipse is the locus of points in a plane, the sum of the distances of each of them 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's denote the foci through the distance between them - through , and a constant value equal to the sum of the distances from each point of the ellipse to the foci, through (by condition ).
Let's build 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 focuses will have the following coordinates: left focus and right focus. Let's derive the equation of the ellipse in the coordinate system we have chosen. To this end, consider an arbitrary point of the ellipse. By definition of an ellipse, the sum of the distances from this point to the foci is:
Using the formula for the distance between two points, we obtain, therefore,
To simplify this equation, we write it in the form
Then squaring both sides of the equation gives
or, after obvious simplifications:
Now again we square both sides of the equation, after which we will have:
or, after identical transformations:
Since according to the condition in the definition of an ellipse , then is a positive number. We introduce the notation
Then the equation will take the following form:
By definition of an ellipse, the coordinates of any of its points satisfy equation (26). But equation (29) is a consequence of equation (26). Therefore, it also satisfies 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's establish the shape of the ellipse using its canonical equation.
First of all, note that this equation contains only even powers of x and y. This means that if any point belongs to an ellipse, then it also includes a point that is symmetrical with a point about the abscissa axis, and a point that is symmetric with a point about the y-axis. Thus, the ellipse has two mutually perpendicular axes of symmetry, which in our chosen coordinate system coincide with the coordinate axes. The axes of symmetry of the ellipse will be called 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's determine the shape of the ellipse first in the first quarter. To do this, we solve equation (28) with respect to y:
It is obvious that here , since y takes imaginary values for . With an 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 conclude 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. It follows from the symmetry of the ellipse that, in addition to the vertices, the ellipse has two more vertices (see Fig. 45).
The segments and connecting the 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 semiaxes 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 , then the eccentricity of the ellipse is less than one: The eccentricity characterizes the shape of the ellipse. Indeed, it follows from formula (28), From this it can be seen that the smaller the eccentricity of the ellipse, the less its minor semiaxis b differs from the major semiaxis a, i.e., the less the ellipse is extended (along the focal axis).
In the limiting case, when you get a circle of radius a: , or . At the same time, the foci of the ellipse, as it were, 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 such an angle a between themselves, for which (Fig. 46). We construct a coordinate system in the P plane, and an Oxy system in the Q plane with a common origin O and a common abscissa axis coinciding with the line of intersection of the planes. Consider in the plane P the circle
centered at the origin and radius a. Let be an arbitrarily chosen point of the circle, be its projection onto the Q plane, and be the projection of the point M onto the Ox axis. Let us show that the point lies on an ellipse with semi-axes a and b.
