Ellipse equation. Canonical Ellipse Equation
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 lines of the second order... The tour has already begun, and from the beginning brief information about the entire exposition on different floors of the museum:
The concept of an algebraic line and its order
A line on a plane is called algebraic, if in affine coordinate system its equation has the form, where is a polynomial consisting of terms of the form (- real number, - non-negative integers).
As you can see, the equation of an algebraic line does not contain sines, cosines, logarithms and other functional beau monde. Only "x" and "games" in non-negative integers 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 assume that all subsequent calculations take place in Cartesian coordinates.
General equation the second-order line has the form, where 
- arbitrary real numbers (it is customary to write with a multiplier - "two"), and the coefficients are not equal to zero at the same time.
If, then the equation is simplified to 
, and if the coefficients are not simultaneously equal to zero, then this is exactly general equation of a "flat" line which is first order line.
Many have 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 order of the line, you need to iterate all terms its equations and for each of them find sum of degrees incoming variables.
For instance:
the term contains "x" in the 1st degree;
the term contains "game" in 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 summand has the sum of the degrees of the variables: 1 + 1 = 2;
the term contains "game" in the 2nd degree;
all other terms - lesser degree.
If we additionally add, say, to our equation, then it will already determine third order line... Obviously, the general form of the third-order line equation contains full set»Terms, the sum of the degrees of the variables in which is equal to three:
, where the coefficients are not equal to zero at the same time.
In the event that we add one or more suitable terms 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 we get acquainted with polar coordinate system.
However, let's return to the general equation and recall its simplest school variations. As examples, a parabola suggests itself, the equation of which can be easily reduced to general view, and a hyperbola with an equivalent equation. However, not everything is so smooth….
Significant disadvantage general equation is that it is almost always not clear which line it sets. Even in the simplest case, you will not immediately realize that this is a hyperbole. Such layouts are good only at a masquerade, therefore, in the course of analytical geometry, it is considered typical task reducing the equation of the second order line to the canonical form.
What is the canonical form of the equation?
It is generally accepted standard view equations, when in a matter of seconds it becomes clear which geometric object it defines. In addition, the canonical view is very convenient for solving many practical assignments... So, for example, according to the canonical equation "Flat" straight, firstly, it is immediately clear that it is a straight line, and secondly, the point belonging to it and the direction vector can be easily seen.
Obviously, any 1st order line is a straight line. On the second floor, however, not a watchman is 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 equation of the second-order line is reduced to one of the following types:
(and are positive real numbers)
1) 
- the canonical equation of the ellipse;
2) - the canonical hyperbole equation;
3) 
- the canonical equation of the parabola;
4) – imaginary ellipse;
5) - a pair of intersecting straight lines;
6) - pair imaginary intersecting lines (with the only valid intersection point at the origin);
7) - a pair of parallel straight lines;
8) - pair imaginary parallel lines;
9) - a pair of coincident straight lines.
Some readers may get the impression that the list is incomplete. For example, in point 7, the equation sets the pair direct parallel to the axis, and the question arises: where is the equation that determines the straight lines parallel to the ordinate? Answer: it not considered canonical... The straight lines represent the same standard case, rotated 90 degrees, and the additional entry in the classification is redundant, since it does not carry anything fundamentally new.
So there are nine and only nine different types lines of the 2nd order, but in practice the most common ellipse, hyperbola and parabola.
Let's look at an 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 of 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 ellipsis”, “the difference between an ellipse and an oval” and “eccentricity of an elebsis”.
The canonical equation of the ellipse has the form, where are positive real numbers, and. I will formulate the very definition of an ellipse later, but for now it's time to take a break from the talking shop and solve a common problem:
How do I build an ellipse?
Yes, take it and just draw it. The task is often encountered, and a significant part of the students do not quite competently cope with the drawing:
Example 1
Construct the ellipse given by the equation
Solution: first we bring the equation to the canonical form: 
Why lead? One of the advantages of the canonical equation is that it allows you to instantly determine ellipse vertices that are in points. It is easy to see that the coordinates of each of these points satisfy the equation.
In this case : 
Section are called major axis ellipse;
section – minor axis;
number 
are called semi-major axis ellipse;
number 
– semi-minor axis.
in our example:.
To quickly imagine what this or that ellipse looks like, it is enough to look at the values "a" and "bs" of its canonical equation.
Everything is fine, foldable and beautiful, but there is one caveat: I made the drawing using the program. And you can complete the drawing with any application. However, in the harsh reality, there is a checkered piece of paper on the table, and mice are dancing in circles on our hands. People with artistic talent, of course, can argue, but you also have mice (though smaller). It's not for nothing that mankind has invented a ruler, compasses, 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 semi-axes. 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 do not like the construction with the help of a compass and a ruler due to not the shortest 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 equation of the ellipse on the draft, quickly express: 
Further, the equation breaks down into two functions: 
- defines the upper arc of the ellipse; 
- defines the lower arc of the ellipse.
The ellipse specified by the canonical equation is symmetric about the coordinate axes, as well as about the origin. And that's great - symmetry is almost always a harbinger of freebies. Obviously, it is enough to deal with the 1st coordinate quarter, so we need the function 
... Finding additional points with abscissas suggests itself 
... We hit three sms on the calculator: 
Of course, it is also pleasant that if allowed serious mistake in calculations, it will immediately become clear during the construction.
Mark the points in the drawing (red), symmetrical points on the remaining 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 give pressure to the pencil. The result should be 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 a philistine sense ("a child drew an oval", etc.). This is a mathematical term that has a detailed formulation. The purpose of this lesson is not to consider the theory of ovals and their various types, which are almost overlooked in the standard course of analytical geometry. And, in line with more relevant needs, we jump straight to the strict definition of an ellipse:
Ellipse Is the set of all points of the plane, the sum of the distances to each of which is 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 focuses is less than this value:.
Now everything will become clearer: 
Imagine that the blue dot is "driving" 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" at the right vertex of the ellipse, then:, which is what you wanted to check.
Another way of drawing it is based on the definition of an ellipse. Higher mathematics, at times, the cause of tension and stress, so it's time to have another unloading session. Please take a Whatman paper or a large piece of cardboard and pin it to the table with two studs. 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 that belongs to the ellipse. Now start tracing your pencil across the sheet of paper, keeping the green thread taut. Continue the process until you return to the starting point ... excellent ... the drawing can be submitted to the teacher for checking =)
How do I find the focuses of an ellipse?
In the given example, I have depicted "ready-made" focal 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 focus to the center of symmetry of the ellipse.
Calculations are easier than a steamed turnip: 
! Concrete coordinates of foci cannot be identified with the meaning "tse"! I repeat that this is DISTANCE from each focus to the center(which in the general case does not have to be located exactly at the origin).
And, therefore, the distance between the foci 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 focuses will naturally change their coordinates. Please consider this moment in the course of further study of the topic.
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 the ellipse depends on its eccentricity. For this fix the left and right vertices the considered ellipse, that is, the value of the semi-major axis will remain constant. Then the eccentricity formula will take the form:.
Let's start bringing the eccentricity value closer to unity. This is only possible if. What does it mean? ... remember the magic tricks 
... This means that the focuses of the ellipse will "move apart" along the abscissa axis to the lateral 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, how closer meaning eccentricity of the ellipse to one, the more elongated the ellipse.
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 "tse" becomes less and less and, accordingly, the eccentricity tends to zero:.
In this case, the “green segments” will, on the contrary, “become crowded” and they will begin to “push” the ellipse line up and down.
In this way, the closer the eccentricity value is to zero, the more the ellipse looks like... look at the extreme case where 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 is reflexively transformed to the well-known from school equation of a circle with a center at the origin of coordinates of radius "a".
In practice, the recording with the "speaking" letter "er" is more often used:. The radius is the length of a segment, with each point of the circle removed from the center by the distance of the radius.
Note that the definition of an ellipse remains completely correct: the focuses coincide, and the sum of the lengths of the coinciding segments for each point of the circle is a constant value. Since the distance between the foci, then the eccentricity of any circle is zero.
A circle is built easily and quickly, it is enough to arm yourself with a compass. Nevertheless, 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 brisk Matan form:
- function of the upper semicircle;
- the function of the lower semicircle.
Then we find desired values, differentiate, integrate and doing other good things.
The article, of course, is for reference only, but how in the world can you live without love? Creative task for independent decision
Example 2
Write the canonical equation of an ellipse if one of its focuses and the semi-minor axis are known (the center is at the origin). Find vertices, additional points and draw a line in the drawing. Calculate eccentricity.
Solution and drawing at the end of the lesson
Let's add an action:
Rotating and parallel translation of 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 examined the ellipse 
, but isn’t in practice 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 really defines an ellipse. Let's dispel mysticism: 
As a result of construction, our native ellipse is obtained, rotated by 90 degrees. That is, 
- it non-canonical notation ellipse 
. Record!- the equation 
does not define any other ellipse, since there are no points (foci) on the axis that would satisfy the definition of an ellipse.
An ellipse is the locus of points of the plane, the sum of the distances from each of which to two given points F_1, and F_2 is a constant value (2a) greater than the distance (2c) between these given points(Figure 3.36, a). This geometric definition expresses focal ellipse property.
Focal property of an ellipse
Points F_1, and F_2 are called focal points of the ellipse, the distance between them is 2c = F_1F_2 - the focal length, the middle O of the segment F_1F_2 - the center of the ellipse, the number 2a - the length of the major axis of the ellipse (respectively, the number a - the semi-major axis of the ellipse). The segments F_1M and F_2M connecting an arbitrary point M of the ellipse with its foci are called the focal radii of the point M. The segment connecting two points of the ellipse is called the chord of the ellipse.
The ratio e = \ frac (c) (a) is called the eccentricity of the ellipse. It follows from the definition (2a> 2c) that 0 \ leqslant e<1 . При e=0 , т.е. при c=0 , фокусы F_1 и F_2 , а также центр O совпадают, и эллипс является окружностью радиуса a (рис.3.36,6).
Geometric definition of an ellipse, which expresses its focal property, is equivalent to its analytical definition - a line defined by the canonical equation of an ellipse:
Indeed, we introduce a rectangular coordinate system (Figure 3.36, c). The center O of the ellipse is taken as the origin of the coordinate system; the straight line passing through the foci (the focal axis or the first axis of the ellipse) is taken as the abscissa axis (the positive direction on it from the point F_1 to the point F_2); the straight line perpendicular to the focal axis and passing through the center of the ellipse (the second axis of the ellipse) is taken as the ordinate (the direction on the ordinate is chosen so that the rectangular coordinate system Oxy is right).
Let's compose the equation of the ellipse, using its geometric definition, which expresses the focal property. In the selected coordinate system, determine the coordinates of the focuses F_1 (-c, 0), ~ F_2 (c, 0)... For an arbitrary point M (x, y) belonging to an ellipse, we have:
\ vline \, \ overrightarrow (F_1M) \, \ vline \, + \ vline \, \ overrightarrow (F_2M) \, \ vline \, = 2a.
Writing this equality in coordinate form, we get:
\ sqrt ((x + c) ^ 2 + y ^ 2) + \ sqrt ((x-c) ^ 2 + y ^ 2) = 2a.
Transfer the second radical to right side, we square both sides of the equation and give similar terms:
(x + c) ^ 2 + y ^ 2 = 4a ^ 2-4a \ sqrt ((xc) ^ 2 + y ^ 2) + (xc) ^ 2 + y ^ 2 ~ \ Leftrightarrow ~ 4a \ sqrt ((xc ) ^ 2 + y ^ 2) = 4a ^ 2-4cx.
Dividing by 4, we square both sides of the equation:
a ^ 2 (xc) ^ 2 + a ^ 2y ^ 2 = a ^ 4-2a ^ 2cx + c ^ 2x ^ 2 ~ \ Leftrightarrow ~ (a ^ 2-c ^ 2) ^ 2x ^ 2 + a ^ 2y ^ 2 = a ^ 2 (a ^ 2-c ^ 2).
By designating b = \ sqrt (a ^ 2-c ^ 2)> 0, we get b ^ 2x ^ 2 + a ^ 2y ^ 2 = a ^ 2b ^ 2... Dividing both sides by a ^ 2b ^ 2 \ ne0, we arrive at the canonical equation of the ellipse:
\ frac (x ^ 2) (a ^ 2) + \ frac (y ^ 2) (b ^ 2) = 1.
Therefore, the selected coordinate system is canonical.
If the foci of the ellipse coincide, then the ellipse is a circle (Figure 3.36.6), since a = b. In this case, any rectangular coordinate system with the origin at the point will be canonical O \ equiv F_1 \ equiv F_2, a equation x ^ 2 + y ^ 2 = a ^ 2 is the equation of a circle centered at O and radius a.
Conducting reasoning in reverse order, it can be shown that all points, the coordinates of which satisfy equation (3.49), and only they, belong to a geometrical place of points called an ellipse. In other words, the analytical definition of an ellipse is equivalent to its geometric definition, which expresses the focal property of the ellipse.
Ellipse directory property
Ellipse directrix are two straight lines that run parallel to the ordinate axis of the canonical coordinate system at the same distance \ frac (a ^ 2) (c) from it. For c = 0, when the ellipse is a circle, there are no directrixes (we can assume that the directrixes are infinitely distant).
Ellipse with eccentricity 0
Indeed, for example, for focus F_2 and directrix d_2 (Fig. 3.37.6), the condition \ frac (r_2) (\ rho_2) = e can be written in coordinate form:
\ sqrt ((x-c) ^ 2 + y ^ 2) = e \ cdot \! \ left (\ frac (a ^ 2) (c) -x \ right)
Getting rid of irrationality and replacing e = \ frac (c) (a), ~ a ^ 2-c ^ 2 = b ^ 2, we arrive at the canonical equation of the ellipse (3.49). Similar reasoning can be carried out for focus F_1 and directrix d_1 \ colon \ frac (r_1) (\ rho_1) = e.
Equation of an ellipse in a polar coordinate system
The equation of the ellipse in the polar coordinate system F_1r \ varphi (Fig. 3.37, c and 3.37 (2)) has the form
r = \ frac (p) (1-e \ cdot \ cos \ varphi)
where p = \ frac (b ^ 2) (a) is the focal parameter of the ellipse.
Indeed, let us choose the left focus F_1 of the ellipse as the pole of the polar coordinate system, and the ray F_1F_2 as the polar axis (Figure 3.37, c). Then, for an arbitrary point M (r, \ varphi), according to the geometric definition (focal property) of an ellipse, we have r + MF_2 = 2a. We express the distance between the points M (r, \ varphi) and F_2 (2c, 0) (see):
\ begin (aligned) F_2M & = \ sqrt ((2c) ^ 2 + r ^ 2-2 \ cdot (2c) \ cdot r \ cos (\ varphi-0)) = \\ & = \ sqrt (r ^ 2- 4 \ cdot c \ cdot r \ cdot \ cos \ varphi + 4 \ cdot c ^ 2). \ End (aligned)
Therefore, in coordinate form, the equation of the ellipse F_1M + F_2M = 2a has the form
r + \ sqrt (r ^ 2-4 \ cdot c \ cdot r \ cdot \ cos \ varphi + 4 \ cdot c ^ 2) = 2 \ cdot a.
We secrete the radical, square both sides of the equation, divide by 4, and give similar terms:
r ^ 2-4 \ cdot c \ cdot r \ cdot \ cos \ varphi + 4 \ cdot c ^ 2 ~ \ Leftrightarrow ~ a \ cdot \! \ left (1- \ frac (c) (a) \ cdot \ cos \ varphi \ right) \! \ cdot r = a ^ 2-c ^ 2.
Express the polar radius r and replace e = \ frac (c) (a), ~ b ^ 2 = a ^ 2-c ^ 2, ~ p = \ frac (b ^ 2) (a):
r = \ frac (a ^ 2-c ^ 2) (a \ cdot (1-e \ cdot \ cos \ varphi)) \ quad \ Leftrightarrow \ quad r = \ frac (b ^ 2) (a \ cdot (1 -e \ cdot \ cos \ varphi)) \ quad \ Leftrightarrow \ quad r = \ frac (p) (1-e \ cdot \ cos \ varphi),
Q.E.D.
The geometric meaning of the coefficients in the ellipse equation
Let's find the points of intersection of the ellipse (see Fig. 3.37, a) with the coordinate axes (the vertices of the zllipse). Substituting y = 0 into the equation, we find the points of intersection of the ellipse with the abscissa axis (with the focal axis): x = \ pm a. Therefore, the length of the segment of the focal axis enclosed within the ellipse is 2a. This segment, as noted above, is called the major axis of the ellipse, and the number a is called the major axis of the ellipse. Substituting x = 0, we get y = \ pm b. Therefore, the length of the segment of the second axis of the ellipse, enclosed within the ellipse, is equal to 2b. This segment is called the minor axis of the ellipse, and the number b is called the minor axis of the ellipse.
Really, b = \ sqrt (a ^ 2-c ^ 2) \ leqslant \ sqrt (a ^ 2) = a, and the equality b = a is obtained only in the case c = 0, when the ellipse is a circle. Attitude k = \ frac (b) (a) \ leqslant1 is called the compression ratio of the ellipse.
Remarks 3.9
1. Straight lines x = \ pm a, ~ y = \ pm b limit on the coordinate plane the main rectangle, inside which there is an ellipse (see Fig. 3.37, a).
2. An ellipse can be defined as locus of points obtained by compressing a circle to its diameter.
Indeed, let in the rectangular coordinate system Oxy the equation of the circle has the form x ^ 2 + y ^ 2 = a ^ 2. When compressed to the abscissa axis with a factor of 0 \ begin (cases) x "= x, \\ y" = k \ cdot y. \ end (cases) Substituting x = x "and y = \ frac (1) (k) y" into the equation of the circle, we obtain the equation for the coordinates of the image M "(x", y ") of the point M (x, y): (x ") ^ 2 + (\ left (\ frac (1) (k) \ cdot y" \ right) \^2=a^2 \quad \Leftrightarrow \quad \frac{(x")^2}{a^2}+\frac{(y")^2}{k^2\cdot a^2}=1 \quad \Leftrightarrow \quad \frac{(x")^2}{a^2}+\frac{(y")^2}{b^2}=1,
!} since b = k \ cdot a. This is the canonical equation of the ellipse. 3. Coordinate axes (canonical coordinate system) are the axes of symmetry of the ellipse (called the principal axes of the ellipse), and its center is the center of symmetry. Indeed, if the point M (x, y) belongs to the ellipse. then the points M "(x, -y) and M" "(- x, y), which are symmetric to the point M with respect to the coordinate axes, also belong to the same ellipse. 4. From the equation of the ellipse in the polar coordinate system r = \ frac (p) (1-e \ cos \ varphi)(see Figure 3.37, c), the geometric meaning of the focal parameter is clarified - this is half the length of the chord of the ellipse passing through its focus perpendicular to the focal axis (r = p at \ varphi = \ frac (\ pi) (2)). 5. Eccentricity e characterizes the shape of an ellipse, namely the difference between an ellipse and a circle. The more e, the more elongated the ellipse, and the closer e is to zero, the closer the ellipse is to the circle (Figure 3.38, a). Indeed, taking into account that e = \ frac (c) (a) and c ^ 2 = a ^ 2-b ^ 2, we obtain e ^ 2 = \ frac (c ^ 2) (a ^ 2) = \ frac (a ^ 2-b ^ 2) (a ^ 2) = 1 - (\ left (\ frac (a) (b) \ right ) \^2=1-k^2,
!} where k is the compression ratio of the ellipse, 0 6. Equation \ frac (x ^ 2) (a ^ 2) + \ frac (y ^ 2) (b ^ 2) = 1 at a 7. Equation \ frac ((x-x_0) ^ 2) (a ^ 2) + \ frac ((y-y_0) ^ 2) (b ^ 2) = 1, ~ a \ geqslant b defines an ellipse centered at the point O "(x_0, y_0), the axes of which are parallel to the coordinate axes (Fig. 3.38, c). This equation is reduced to the canonical one using parallel translation (3.36). 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 Ellipse Equation 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 ellipse \ frac (x ^ 2) (2 ^ 2) + \ frac (y ^ 2) (1 ^ 2) = 1 in the canonical coordinate system Oxy. Find semiaxes, focal length, eccentricity, compression ratio, focal parameter, directrix equations. Solution. Comparing the given equation with the canonical one, we determine the semiaxes: a = 2 - semi-major axis, b = 1 - semi-minor axis of the ellipse. We build the main rectangle with sides 2a = 4, ~ 2b = 2 centered at the origin (Figure 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 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)). The canonical equation of the ellipse has the form where a is the semi-major axis; b - semi-minor axis. Points F1 (c, 0) and F2 (-c, 0) - c are called a, b - semiaxes of the ellipse. Finding foci, eccentricity, directrix of an ellipse, if its canonical equation is known. Definition of hyperbole. Focuses of hyperbole. Definition. A hyperbola is a set of points on a plane for which the modulus of the difference between the distances from two given points, called foci, is a constant value less than the distance between the foci. By definition, | r1 - r2 | = 2a. F1, F2 - hyperbolic foci. F1F2 = 2c. Canonical equation of hyperbola. Semi-axes of the hyperbola. Construction of a hyperbola if its canonical equation is known. Canonical equation: The semimajor axis of the hyperbola is half the minimum distance between the two branches of the hyperbola, on the positive and negative sides of the axis (left and right relative to the origin). For a branch located on the positive side, the semiaxis will be equal to: If we express it through a conical section and eccentricity, then the expression will take the form: Finding foci, eccentricity, directrix of hyperbola, if its canonical equation is known. Eccentricity of hyperbola Definition. The ratio is called the eccentricity of the hyperbola, where c - half the distance between the foci, and is the real semiaxis. Considering that c2 - a2 = b2: If a = b, e =, then the hyperbola is called isosceles (equilateral). Hyperbole directrixes Definition. Two straight lines perpendicular to the real axis of the hyperbola and located symmetrically about the center at a distance a / e from it are called directrix of the hyperbola. Their equations are:. Theorem. If r is the distance from an arbitrary point M of the hyperbola to any focus, d is the distance from the same point to the directrix corresponding to this focus, then the ratio r / d is a constant value equal to the eccentricity. Definition of a parabola. Focus and headmistress of the parabola. Parabola. A parabola is a locus of points, each of which is equally distant from a given fixed point and from a given fixed straight line. The point referred to in the definition is called the focus of the parabola, and the line is called its directrix. Canonical equation of a parabola. Parabola parameter. Construction of a parabola. The canonical equation of a parabola in a rectangular coordinate system: (or, if the axes are swapped). The construction of the parabola for a given value of the parameter p is performed in the following sequence: The axis of symmetry of the parabola is drawn and the segment KF = p is laid on it; Directrix DD1 is drawn through point K perpendicular to the axis of symmetry; The segment KF is divided in half; the vertex 0 of the parabola is obtained; A number of arbitrary points 1, 2, 3, 5, 6 are measured from the top with a gradually increasing distance between them; Through these points, auxiliary straight lines are drawn perpendicular to the axis of the parabola; On auxiliary lines, serifs are made with a radius equal to the distance from the line to the directrix; The resulting points are connected by a smooth curve. 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 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, using two needles. It is clear that this is possible only for a ≥ c. Stretching 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. For a = c, the ellipse is a segment with ends F 1 and F 2, and for 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 the definition 7.1 of the ellipse (see Fig. 7.1) are called foci of an ellipse, the distance between them, denoted by 2c, is focal distance, and the segments F 1 M and F 2 M connecting an arbitrary point M on the ellipse with its foci are focal radii. The shape of the ellipse is completely determined by the focal distance | F 1 F 2 | = 2с and parameter a, and its position on the plane is a pair of points F 1 and F 2. From the definition of an ellipse it follows that it is symmetric with respect to the straight line passing through the foci F 1 and F 2, as well as with respect to the 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) is its semi-minor axis... It is easy to see that for c> 0, the semi-major axis a is equal to the distance from the center of the ellipse to those of its vertices that are on the same axis with the focal points of the ellipse (vertices A and B in Fig. 7.2, a), and the semi-minor axis 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 on the plane some ellipse with foci at points F 1 and F 2, the major axis 2a. Let 2c be the focal distance, 2c = | F 1 F 2 | Let us 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 axis(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, the foci have coordinates F 1 (c; 0), F 2 (-c; 0). Using the formula for the distance between points, we write down 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. Therefore, we transform it. Move the second radical in equation (7.2) to the right-hand 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 similar terms, we obtain √ ((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, taking into account the value of the introduced 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 squaring, removing square radicals. Squaring an equation is an equivalent transformation if both sides contain values with the same sign, but we did not check this in our transformations. We may not check the equivalence of transformations if we take into account 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 particular 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 the set of points whose coordinates satisfy Eq. (7.4) with a given value of the parameter a. Can this set be distributed among several ellipses? Some points of the set belong to an ellipse with semi-major axis a. Let this set contain a point lying on an ellipse with 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 see that the system has no solutions for г ≠ a. To do this, it is enough to exclude, for example, x from the first equation: which after transformations leads to the equation which has no solutions for г ≠ a, since. So, (7.4) is the equation of an ellipse with semi-major axis a> 0 and semi-minor axis b = √ (a 2 - c 2)> 0. It is called the canonical ellipse equation. Ellipse view. The above geometric method for constructing an ellipse gives a sufficient idea of appearance ellipse. But the form of the ellipse can also be investigated with the help of its canonical equation (7.4). For example, assuming y ≥ 0, we can express y in terms of x: y = b√ (1 - x 2 / a 2), and after examining this function, build its graph. There is another way to build 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 we compress it with a coefficient a / b> 1 along ordinate axes, then you get a curve that is described by the equation x 2 + (ya / b) 2 = a 2, that is, an ellipse. Remark 7.1. If the same circle is compressed with the a / b ratio Ellipse eccentricity... The ratio of the focal distance of an ellipse to its major axis is called eccentricity of the ellipse and are denoted by ε. For an ellipse given canonical equation (7.4), ε = 2c / 2a = c / 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), since equations (7.4) and (7.2) are equivalent. Therefore, the equation of the ellipse is also (7.3). 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 considerations of symmetry or by repeating calculations in which the first radical is transferred to the right-hand side, and not the second, before the square of equation (7.2). 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 equation for an ellipse. Example 7.1. Let's find the canonical equation of an ellipse with a semi-major axis of 5 and an eccentricity of 0.8 and construct it. Knowing the semi-major axis of the ellipse a = 5 and the eccentricity ε = 0.8, we find its semi-minor axis b. Since b = √ (a 2 - c 2), and c = εa = 4, then b = √ (5 2 - 4 2) = 3. Hence the canonical equation has the form x 2/5 2 + y 2/3 2 = 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 symmetry axes 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 into it. In fig. 7.4, the foci of the F 1,2 (± 4; 0) ellipse are also indicated. Geometric properties of the ellipse. We rewrite the first equation in (7.6) as | F 1 M | = (a / ε - x) ε. Note that the quantity 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 straight line d: x = a / ε from the point M (x; y) lying to the left of this straight line. The ellipse equation can be written as | F 1 M | / (a / 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 straight line d has a "twin" - the vertical line d ", symmetric to d about the center of the ellipse, which is given by the equation x = -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 directrix... The directrixes of the ellipse are perpendicular to the axis of symmetry of the ellipse, on which its foci are located, and are spaced from the center of the ellipse at a distance a / ε = a 2 / c (see Fig. 7.5). The distance p from the directrix to the nearest focus to it is called focal parameter of the ellipse... This parameter is p = a / ε - c = (a 2 - c 2) / c = 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 in 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 rays coming out of focus F 1 will be concentrated in the second focus F 2, and vice versa. Based on this interpretation, the specified 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 ellipse, hyperbola, and parabola. The general view of the equation of the second-order curve is as follows: where A, B, C, D, E, F- numbers and at least one of the coefficients A, B, C is not zero. When solving problems with curves of the second order, 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. Definition of an ellipse. An ellipse is the set of all points of the plane, such for which the sum of the distances to the points, called foci, is a constant value and greater than the distance between the foci. Focuses are indicated as in the figure below. The canonical equation of the ellipse is: where a and b (a > b) - the lengths of the semiaxes, that is, 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 a 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 intersects the ellipse at points ( a, O) and (- a, O), and the ordinate axis is at the 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 = b, then the equation of the ellipse takes the form. This is the equation of a circle of radius a, and a circle is a special case of an ellipse. An ellipse can be obtained from a circle of radius a if you compress it into a/b times along the axis Oy
. Example 1. Check if the line given by the general equation Solution. We make transformations of the general equation. We apply the transfer of the free term to the right side, term-by-term division of the equation by the same number and 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 major semiaxis 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 are called tricks. called eccentricity ellipse. Attitude b/a characterizes the "flattening" 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 in terms of eccentricity, the formula for 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 equal to 10, then its half, that is, the semiaxis a = 5
, If the distance between the foci is 8, then the number c of 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. 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 equation of eccentricity, we express the number c required 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. Find the number c defining the first coordinates of the focuses of the ellipse: We get the focuses of the ellipse: Example 6. Ellipse foci are located on the axis Ox symmetric 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 focuses is at the point (6; 0) If is an arbitrary point of the ellipse (in the drawing it is indicated in green in the upper right part of the ellipse) and is the distance to this point from the focuses, 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 are called directors ellipse (in the drawing - red lines at the edges). From the two above equations it follows that for any point of the ellipse where and are the distances of this point to the directrix and. Example 7. An ellipse is given. Make an equation for its directors. Solution. We look at the directrix equation and find that it is required to find the eccentricity of the ellipse, i.e. All the data for this is there. We calculate: We get the equation for the directrix of the ellipse: Example 8. Write the canonical equation of an ellipse if its focuses are points and directrixes are straight lines.Parametric Ellipse Equation
Ellipse given by the canonical equation
, an ellipse.
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We continue to solve problems on the ellipse together
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