Class 10 . Curves of the second order.
10.1. Ellipse. Canonical equation. Half shafts, eccentricity, graph.
10.2. Hyperbola. Canonical equation. Semiaxes, eccentricity, asymptotes, graph.
10.3. Parabola. Canonical equation. Parabola parameter, graph.
Curves of the second order in the plane are called lines, the implicit specification of which has the form:
where 
- given real numbers, 
- coordinates of curve points. The most important lines among curves of the second order are ellipse, hyperbola, parabola.
10.1. Ellipse. Canonical equation. Half shafts, eccentricity, graph.
Definition of an ellipse.An ellipse is a plane curve whose sum of distances from two fixed points 
plane to any point 
(those.). points 
called the foci of the ellipse.
Canonical equation of an ellipse:
.
(2)
(or axis 
) passes through foci 
, and the origin is a point 
- located in the center of the segment 
(Fig. 1). Ellipse (2) is symmetrical with respect to the coordinate axes and the origin (the center of the ellipse). Permanent 
,
called semi-axes of an ellipse.
If the ellipse is given by equation (2), then the foci of the ellipse are found as follows.
1) First, we determine where the foci lie: the foci lie on the coordinate axis on which the major semiaxes are located.
2) Then the focal length is calculated 
(distance from foci to origin).
At 
focuses lie on the axis 
;
;
.
At 
focuses lie on the axis 
;
;
.
eccentricity ellipse is called the value: 
(at 
);
(at 
).
Ellipse always has 
. The eccentricity is a characteristic of the compression of the ellipse.
If the ellipse (2) is moved so that the center of the ellipse is at the point 
,
, then the equation of the resulting ellipse has the form
.
10.2. Hyperbola. Canonical equation. Semiaxes, eccentricity, asymptotes, graph.
Definition of a hyperbola.A hyperbola is a plane curve, in which the absolute value of the difference in distances from two fixed points 
plane to any point 
this curve is a constant independent of the point 
(those.). points 
called the foci of the hyperbola.
Canonical equation of a hyperbola:
or 
.
(3)
Such an equation is obtained if the coordinate axis 
(or axis 
) passes through foci 
, and the origin is a point 
- located in the center of the segment 
. Hyperbolas (3) are symmetrical with respect to the coordinate axes and the origin. Permanent 
,
called semiaxes of the hyperbola.
The foci of the hyperbola are found as follows.
At the hyperbole 
focuses lie on the axis 
:
(Fig. 2.a).
At the hyperbole 
focuses lie on the axis 
:
(Fig. 2.b)
Here 
- focal length (distance from the foci to the origin). It is calculated by the formula: 
.
eccentricity hyperbola is called the value:
(for 
);
(for 
).
Hyperbole always has 
.
Asymptotes of hyperbolas(3) are two straight lines: 
. Both branches of the hyperbola approach the asymptotes indefinitely as 
.
The construction of a graph of a hyperbola should be carried out as follows: first, along the semiaxes 
we build an auxiliary rectangle with sides parallel to the coordinate axes; then we draw straight lines through the opposite vertices of this rectangle, these are the asymptotes of the hyperbola; finally, we depict the branches of the hyperbola, they touch the midpoints of the corresponding sides of the auxiliary rectangle and approach with growth 
to asymptotes (Fig. 2).
If the hyperbolas (3) are moved so that their center falls on the point 
, and the semiaxes will remain parallel to the axes 
,
, then the equation of the resulting hyperbolas can be written in the form
,
.
10.3. Parabola. Canonical equation. Parabola parameter, graph.
Definition of a parabola.A parabola is a plane curve in which for any point 
this curve is the distance from 
to a fixed point 
plane (called the focus of the parabola) is equal to the distance from 
to a fixed line on the plane(called the directrix of the parabola) .
Canonical parabola equation:
,
(4)
where 
is a constant called parameter parabolas.
Dot 
parabola (4) is called the vertex of the parabola. Axis 
is the axis of symmetry. The focus of the parabola (4) is at the point 
, directrix equation 
. Parabola plots (4) with values 
and 
shown in fig. 3.a and 3.b, respectively.
The equation 
also defines a parabola in the plane 
, which, compared with parabola (4), has axes 
,
switched places.
If the parabola (4) is moved so that its vertex hits the point 
, and the axis of symmetry will remain parallel to the axis 
, then the equation of the resulting parabola has the form
.
Let's move on to examples.
Example 1. The second order curve is given by the equation 
. Give a name to this curve. Find its foci and eccentricity. Draw a curve and its foci in a plane 
.
Decision. This curve is an ellipse centered at the point 
and axle shafts 
. This can be easily verified by replacing 
. This transformation means moving from a given Cartesian coordinate system 
to the new Cartesian coordinate system 
, whose axes 
parallel to the axes 
,
. This coordinate transformation is called a system shift. 
exactly 
. In the new coordinate system 
the equation of the curve is converted to the canonical equation of the ellipse 
, its graph is shown in Fig. 4.
Let's find tricks. 
, so the tricks 
ellipse located on the axis 
.. In the coordinate system 
:
. Because 
, in the old coordinate system 
focuses have coordinates.
Example 2. Give the name of the curve of the second order and give its graph.
Decision. We select full squares by terms containing variables 
and 
.
Now, the curve equation can be rewritten as:
Therefore, the given curve is an ellipse centered at the point 
and axle shafts 
. The information obtained allows us to draw its graph.
Example 3. Give a name and draw a line graph 
.
Decision. . This is the canonical equation of an ellipse centered at a point 
and axle shafts 
.
Insofar as, 
, we conclude: the given equation defines on the plane 
the lower half of the ellipse (Fig. 5).
Example 4. Give the name of the curve of the second order 
. Find her tricks, eccentricity. Give a graph of this curve.
- canonical equation of a hyperbola with semiaxes 
.
Focal length.
The minus sign is in front of the term with 
, so the tricks 
hyperbolas lie on the axis 
:. The branches of the hyperbola are located above and below the axis 
.
is the eccentricity of the hyperbola.
Asymptotes of a hyperbola: .
The construction of a graph of this hyperbola is carried out in accordance with the above procedure: we build an auxiliary rectangle, draw the asymptotes of the hyperbola, draw the branches of the hyperbola (see Fig. 2.b).
Example 5. Find out the form of the curve given by the equation 
and plot it.
- hyperbola centered at a point 
and half shafts.
Because , we conclude: the given equation determines the part of the hyperbola that lies to the right of the line 
. It is better to draw a hyperbola in an auxiliary coordinate system 
obtained from the coordinate system 
shift 
, and then with a thick line select the desired part of the hyperbola
Example 6. Find out the type of curve and draw its graph.
Decision. Select the full square by the terms with the variable 
:
Let's rewrite the equation of the curve.
This is the equation of a parabola with vertex at the point 
. By a shift transformation, the parabola equation is reduced to the canonical form 
, from which it can be seen that is the parameter of the parabola. Focus 
parabolas in the system 
has coordinates 
,, and in the system 
(according to the shift transformation). The parabola graph is shown in fig. 7.
Homework.
1. Draw ellipses given by the equations: 
Find their semiaxes, focal length, eccentricity and indicate on the ellipse graphs the locations of their foci.
2. Draw hyperbolas given by the equations: 
Find their semi-axes, focal length, eccentricity and indicate on the graphs of hyperbolas the location of their foci. Write the equations for the asymptotes of the given hyperbolas.
3. Draw the parabolas given by the equations: 
. Find their parameter, focal length and indicate the location of the focus on the parabola graphs.
4. Equation 
defines a part of the curve of the 2nd order. Find the canonical equation of this curve, write down its name, build its graph and highlight on it that part of the curve that corresponds to the original equation.
A hyperbola is a locus of points in a plane, the modulus of the difference in distances from each of which to two given points F_1 and F_2 is a constant value (2a), less than the distance (2c) between these given points (Fig. 3.40, a). This geometric definition expresses focal property of a hyperbola.
Focal property of a hyperbola
The points F_1 and F_2 are called the foci of the hyperbola, the distance 2c=F_1F_2 between them is the focal length, the midpoint O of the segment F_1F_2 is the center of the hyperbola, the number 2a is the length of the real axis of the hyperbola (respectively, a is the real semiaxis of the hyperbola). The segments F_1M and F_2M connecting an arbitrary point M of the hyperbola with its foci are called the focal radii of the point M . A line segment connecting two points of a hyperbola is called a chord of the hyperbola.
The relation e=\frac(c)(a) , where c=\sqrt(a^2+b^2) , is called hyperbolic eccentricity. From the definition (2a<2c) следует, что e>1 .
Geometric definition of a hyperbola, expressing its focal property, is equivalent to its analytical definition - the line given by the canonical equation of the hyperbola:
\frac(x^2)(a^2)-\frac(y^2)(b^2)=1.
Indeed, let's introduce a rectangular coordinate system (Fig. 3.40, b). We take the center O of the hyperbola as the origin of the coordinate system; the straight line passing through the foci (focal axis), we will take as the abscissa axis (the positive direction on it from the point F_1 to the point F_2); a straight line perpendicular to the abscissa axis and passing through the center of the hyperbola is taken as the ordinate axis (the direction on the ordinate axis is chosen so that the rectangular coordinate system Oxy is right).
Let's write the equation of the hyperbola using the geometric definition expressing the focal property. In the selected coordinate system, we determine the coordinates of the foci F_1(-c,0) and F_2(c,0) . For an arbitrary point M(x,y) belonging to a hyperbola, we have:
\left||\overrightarrow(F_1M)|-|\overrightarrow(F_2M)|\right|=2a.
Writing this equation in coordinate form, we get:
\sqrt((x+c)^2+y^2)-\sqrt((x-c)^2+y^2)=\pm2a.
Performing transformations similar to those used in the derivation of the ellipse equation (i.e. getting rid of irrationality), we arrive at the canonical equation of the hyperbola:
\frac(x^2)(a^2)-\frac(y^2)(b^2)=1\,
where b=\sqrt(c^2-a^2) , i.e. the chosen coordinate system is canonical.
By reasoning backwards, it can be shown that all the points whose coordinates satisfy equation (3.50), and only they, belong to the locus of points, called the hyperbola. Thus, the analytic definition of a hyperbola is equivalent to its geometric definition.
Directory property of a hyperbola
Directrixes of a hyperbola are called two straight lines passing parallel to the y-axis of the canonical coordinate system at the same distance a^2\!\!\not(\phantom(|))\,c from it (Fig. 3.41, a). For a=0 , when the hyperbola degenerates into a pair of intersecting lines, the directrixes coincide.
A hyperbola with eccentricity e=1 can be defined as the locus of points in the plane, for each of which the ratio of the distance to a given point F (focus) to the distance to a given straight line d (directrix) that does not pass through a given point is constant and equal to the eccentricity e ( directory property of a hyperbola). Here F and d are one of the foci of the hyperbola and one of its directrixes, located on the same side of the y-axis of the canonical coordinate system.
Indeed, for example, for the focus F_2 and the directrix d_2 (Fig. 3.41, a) the condition \frac(r_2)(\rho_2)=e can be written in coordinate form:
\sqrt((x-c)^2+y^2)=e\left(x-\frac(a^2)(c)\right)
Getting rid of irrationality and replacing e=\frac(c)(a),~c^2-a^2=b^2, we arrive at the canonical equation of the hyperbola (3.50). Similar reasoning can be carried out for the focus F_1 and the directrix d_1 :
\frac(r_1)(\rho_1)=e \quad \Leftrightarrow \quad \sqrt((x+c)^2+y^2)= e\left(x+\frac(a^2)(c) \right ).
Hyperbola equation in polar coordinates
The equation of the right branch of the hyperbola in the polar coordinate system F_2r\varphi (Fig. 3.41, b) has the form
r=\frac(p)(1-e\cdot\cos\varphi), where p=\frac(p^2)(a) - hyperbola focal parameter.
Indeed, let's choose the right focus F_2 of the hyperbola as the pole of the polar coordinate system, and the ray with the origin at the point F_2, belonging to the line F_1F_2, but not containing the point F_1 (Fig. 3.41, b) as the polar axis. Then for an arbitrary point M(r,\varphi) belonging to the right branch of the hyperbola, according to the geometric definition (focal property) of the hyperbola, we have F_1M-r=2a . We express the distance between the points M(r,\varphi) and F_1(2c,\pi) (see point 2 of remarks 2.8):
F_1M=\sqrt((2c)^2+r^2-2\cdot(2c)^2\cdot r\cdot\cos(\varphi-\pi))=\sqrt(r^2+4\cdot c \cdot r\cdot\cos\varphi+4\cdot c^2).
Therefore, in coordinate form, the equation of a hyperbola has the form
\sqrt(r^2+4\cdot c\cdot r\cdot\cos\varphi+4\cdot c^2)-r=2a.
We isolate the radical, square both sides of the equation, divide by 4 and give like terms:
r^2+4cr\cdot\cos\varphi+4c^2=4a^2+4ar+r^2 \quad \Leftrightarrow \quad a\left(1-\frac(c)(a)\cos\varphi\ right)r=c^2-a^2.
We express the polar radius r and make substitutions e=\frac(c)(a),~b^2=c^2-a^2,~p=\frac(b^2)(a):
r=\frac(c^2-a^2)(a(1-e\cos\varphi)) \quad \Leftrightarrow \quad r=\frac(b^2)(a(1-e\cos\varphi )) \quad \Leftrightarrow \quad r=\frac(p)(1-e\cos\varphi),
Q.E.D. Note that in polar coordinates, the hyperbola and ellipse equations coincide, but describe different lines, since they differ in eccentricities (e>1 for a hyperbola, 0\leqslant e<1 для эллипса).
The geometric meaning of the coefficients in the hyperbola equation
Let's find the intersection points of the hyperbola (Fig. 3.42, a) with the abscissa axis (vertices of the hyperbola). Substituting y=0 into the equation, we find the abscissas of the intersection points: x=\pm a . Therefore, the vertices have coordinates (-a,0),\,(a,0) . The length of the segment connecting the vertices is 2a . This segment is called the real axis of the hyperbola, and the number a is the real semiaxis of the hyperbola. Substituting x=0 , we get y=\pm ib . The length of the segment of the y-axis connecting the points (0,-b),\,(0,b) is equal to 2b . This segment is called the imaginary axis of the hyperbola, and the number b is called the imaginary semiaxis of the hyperbola. The hyperbola intersects the line containing the real axis and does not intersect the line containing the imaginary axis.
Remarks 3.10.
1. The lines x=\pm a,~y=\pm b limit the main rectangle on the coordinate plane, outside of which the hyperbola is located (Fig. 3.42, a).
2. Straight lines containing the diagonals of the main rectangle are called asymptotes of the hyperbola (Fig. 3.42, a).
For equilateral hyperbola, described by the equation (i.e. with a=b ), the main rectangle is a square, the diagonals of which are perpendicular. Therefore, the asymptotes of an equilateral hyperbola are also perpendicular, and they can be taken as the coordinate axes of the rectangular coordinate system Ox"y" (Fig. 3.42, b). In this coordinate system, the hyperbola equation has the form y"=\frac(a^2)(2x")(the hyperbola coincides with the graph of an elementary function expressing an inversely proportional relationship).
Indeed, let us rotate the canonical coordinate system by the angle \varphi=-\frac(\pi)(4)(Fig. 3.42, b). In this case, the coordinates of the point in the old and new coordinate systems are related by the equalities
\left\(\!\begin(aligned)x&=\frac(\sqrt(2))(2)\cdot x"+\frac(\sqrt(2))(2)\cdot y",\\ y& =-\frac(\sqrt(2))(2)\cdot x"+\frac(\sqrt(2))(2)\cdot y"\end(aligned)\right. \quad \Leftrightarrow \quad \ left\(\!\begin(aligned)x&=\frac(\sqrt(2))(2)\cdot(x"+y"),\\ y&=\frac(\sqrt(2))(2) \cdot(y"-x")\end(aligned)\right.
Substituting these expressions into the equation \frac(x^2)(a^2)-\frac(y^2)(a^2)=1 of an equilateral hyperbola and bringing like terms, we obtain
\frac(\frac(1)(2)(x"+y")^2)(a^2)-\frac(\frac(1)(2)(y"-x")^2)(a ^2)=1 \quad \Leftrightarrow \quad 2\cdot x"\cdot y"=a^2 \quad \Leftrightarrow \quad y"=\frac(a^2)(2\cdot x").
3. The coordinate axes (of the canonical coordinate system) are the axes of symmetry of the hyperbola (called the main axes of the hyperbola), and its center is the center of symmetry.
Indeed, if the point M(x,y) belongs to the hyperbola . then the points M"(x,y) and M""(-x,y) , symmetrical to the point M with respect to the coordinate axes, also belong to the same hyperbola.
The axis of symmetry, on which the foci of the hyperbola are located, is the focal axis.
4. From the hyperbola equation in polar coordinates r=\frac(p)(1-e\cos\varphi)(see Fig. 3.41, b) the geometric meaning of the focal parameter is clarified - this is half the length of the chord of the hyperbola passing through its focus perpendicular to the focal axis (r = p at \varphi=\frac(\pi)(2)).
5. The eccentricity e characterizes the shape of the hyperbola. The more e, the wider the branches of the hyperbola, and the closer e is to one, the narrower the branches of the hyperbola (Fig. 3.43, a).
Indeed, the value \gamma of the angle between the asymptotes of the hyperbola containing its branch is determined by the ratio of the sides of the main rectangle: \operatorname(tg)\frac(\gamma)(2)=\frac(b)(2). Considering that e=\frac(c)(a) and c^2=a^2+b^2 , we get
e^2=\frac(c^2)(a^2)=\frac(a^2+b^2)(a^2)=1+(\left(\frac(b)(a)\right )\^2=1+\operatorname{tg}^2\frac{\gamma}{2}. !}
The larger e , the larger the \gamma angle. For an equilateral hyperbola (a=b) we have e=\sqrt(2) and \gamma=\frac(\pi)(2). For e>\sqrt(2) the angle \gamma is obtuse, but for 1 6. Two hyperbolas defined in the same coordinate system by the equations \frac(x^2)(a^2)-\frac(y^2)(b^2)=1 and are called linked to each other. Conjugate hyperbolas have the same asymptotes (Fig. 3.43, b). Conjugate Hyperbola Equation -\frac(x^2)(a^2)+\frac(y^2)(b^2)=1 is reduced to the canonical one by renaming the coordinate axes (3.38). 7. Equation \frac((x-x_0)^2)(a^2)-\frac((y-y_0)^2)(b^2)=1 defines a hyperbola centered at the point O "(x_0, y_0) , whose axes are parallel to the coordinate axes (Fig. 3.43, c). This equation is reduced to the canonical one using parallel translation (3.36). Equation -\frac((x-x_0)^2)(a^2)+\frac((y-y_0)^2)(b^2)=1 defines a conjugate hyperbola centered at the point O"(x_0,y_0) . The parametric equation of a hyperbola in the canonical coordinate system has the form \begin(cases)x=a\cdot\operatorname(ch)t,\\y=b\cdot\operatorname(sh)t,\end(cases)t\in\mathbb(R), where \operatorname(ch)t=\frac(e^t+e^(-t))(2)- hyperbolic cosine, a \operatorname(sh)t=\frac(e^t-e^(-t))(2) hyperbolic sine. Indeed, substituting the coordinate expressions into equation (3.50), we arrive at the main hyperbolic identity \operatorname(ch)^2t-\operatorname(sh)^2t=1. Example 3.21. Draw a hyperbole \frac(x^2)(2^2)-\frac(y^2)(3^2)=1 in the canonical coordinate system Oxy . Find semiaxes, focal length, eccentricity, focal parameter, equations of asymptotes and directrixes. Decision. Comparing the given equation with the canonical one, we determine the semiaxes: a=2 - real semiaxis, b=3 - imaginary semiaxis of the hyperbola. We build the main rectangle with sides 2a=4,~2b=6 centered at the origin (Fig.3.44). We draw asymptotes by extending the diagonals of the main rectangle. We build a hyperbola, taking into account its symmetry about the coordinate axes. If necessary, we determine the coordinates of some points of the hyperbola. For example, substituting x=4 into the hyperbola equation, we get \frac(4^2)(2^2)-\frac(y^2)(3^2)=1 \quad \Leftrightarrow \quad y^2=27 \quad \Leftrightarrow \quad y=\pm3\sqrt (3). Therefore, the points with coordinates (4;3\sqrt(3)) and (4;-3\sqrt(3)) belong to the hyperbola. Calculating focal length 2\cdot c=2\cdot\sqrt(a^2+b^2)=2\cdot\sqrt(2^2+3^2)=2\sqrt(13) eccentricity e=\frac(c)(a)=\frac(\sqrt(13))(2); focal parameter p=\frac(b^2)(a)=\frac(3^2)(2)=4,\!5. We compose the equations of asymptotes y=\pm\frac(b)(a)\,x, i.e y=\pm\frac(3)(2)\,x, and directrix equations: x=\pm\frac(a^2)(c)=\frac(4)(\sqrt(13)). For the rest of the readers, I propose to significantly replenish their school knowledge about parabola and hyperbola. Hyperbola and parabola - is it simple? … Don't wait =) The general structure of the presentation of the material will resemble the previous paragraph. Let's start with the general concept of a hyperbola and the problem of its construction. The canonical equation of a hyperbola has the form , where are positive real numbers. Note that, unlike ellipse, the condition is not imposed here, that is, the value of "a" may be less than the value of "be". I must say, quite unexpectedly ... the equation of the "school" hyperbole does not even closely resemble the canonical record. But this riddle will still have to wait for us, but for now let's scratch the back of our head and remember what characteristic features the curve under consideration has? Let's spread it on the screen of our imagination function graph …. A hyperbola has two symmetrical branches. Good progress! Any hyperbole has these properties, and now we will look with genuine admiration at the neckline of this line: Example 4 Construct a hyperbola given by the equation Decision: at the first step, we bring this equation to the canonical form . Please remember the typical procedure. On the right, you need to get a “one”, so we divide both parts of the original equation by 20: Here you can reduce both fractions, but it is more optimal to make each of them three-story: And only after that to carry out the reduction: We select the squares in the denominators: Why is it better to carry out transformations in this way? After all, the fractions of the left side can be immediately reduced and get. The fact is that in the example under consideration, we were a little lucky: the number 20 is divisible by both 4 and 5. In the general case, such a number does not work. Consider, for example, the equation . Here, with divisibility, everything is sadder and without three-story fractions no longer needed: So, let's use the fruit of our labors - the canonical equation: There are two approaches to constructing a hyperbola - geometric and algebraic. It is advisable to adhere to the following algorithm, first the finished drawing, then the comments: In practice, a combination of rotation through an arbitrary angle and parallel translation of a hyperbola is often encountered. This situation is discussed in the lesson. Reduction of the 2nd order line equation to the canonical form. It's done! She is the most. Ready to reveal many secrets. The canonical equation of a parabola has the form , where is a real number. It is easy to see that in its standard position the parabola "lies on its side" and its vertex is at the origin. In this case, the function sets the upper branch of this line, and the function sets the lower branch. Obviously, the parabola is symmetrical about the axis. Actually, what to bathe: Example 6 Build a parabola Decision: the vertex is known, let's find additional points. The equation In order to shorten the record, we will carry out calculations “under the same brush”: For compact notation, the results could be summarized in a table. Before performing an elementary point-by-point drawing, we formulate a strict A parabola is the set of all points in a plane that are equidistant from a given point and a given line that does not pass through the point. The point is called focus parabolas, straight line headmistress (written with one "es") parabolas. The constant "pe" of the canonical equation is called focal parameter, which is equal to the distance from the focus to the directrix. AT this case. In this case, the focus has coordinates , and the directrix is given by the equation . Congratulations! Many of you have made a real discovery today. It turns out that the hyperbola and parabola are not at all graphs of "ordinary" functions, but have a pronounced geometric origin. Obviously, with an increase in the focal parameter, the branches of the graph will “spread out” up and down, approaching the axis infinitely close. With a decrease in the value of "pe", they will begin to shrink and stretch along the axis The eccentricity of any parabola is equal to one: The parabola is one of the most common lines in mathematics, and you will have to build it really often. Therefore, please pay special attention to the final paragraph of the lesson, where I will analyze the typical options for the location of this curve. ! Note
: as in the cases with the previous curves, it is more correct to talk about the rotation and parallel translation of the coordinate axes, but the author will limit himself to a simplified version of the presentation so that the reader has an elementary idea of \u200b\u200bthese transformations. Definition. The hyperbola is the locus of points in the plane y, the absolute value of the difference in the distances of each of which from two given points of this plane, called foci, y has a constant value, provided that this value is not equal to zero and is less than the distance between the foci. Let us denote the distance between the foci as a constant value equal to the modulus of the difference in distances from each point of the hyperbola to the foci, through (by condition ). As in the case of an ellipse, we draw the abscissa axis through the foci, and take the middle of the segment as the origin (see Fig. 44). Foci in such a system will have coordinates Let us derive the equation of the hyperbola in the chosen coordinate system. By definition of a hyperbola, for any of its points we have or But . Therefore, we get After simplifications similar to those made when deriving the ellipse equation, we get the following equation: which is a consequence of equation (33). It is easy to see that this equation coincides with equation (27) obtained for an ellipse. However, in equation (34) the difference , since for the hyperbola . Therefore, we put Then equation (34) is reduced to the following form: This equation is called the canonical equation of the hyperbola. Equation (36), as a consequence of equation (33), is satisfied by the coordinates of any point of the hyperbola. It can be shown that the coordinates of points that do not lie on the hyperbola do not satisfy equation (36). Let us establish the form of the hyperbola using its canonical equation. This equation contains only even powers of the current coordinates. Consequently, the hyperbola has two axes of symmetry, in this case coinciding with the coordinate axes. In what follows, the axes of symmetry of the hyperbola will be called the axes of the hyperbola, and the point of their intersection will be called the center of the hyperbola. The axis of the hyperbola on which the foci are located is called the focal axis. We explore the shape of the hyperbola in the first quarter, where Here, because otherwise y would take imaginary values. As x increases from a to, it increases from 0 to The part of the hyperbola lying in the first quarter will be the arc shown in Fig. 47. Since the hyperbola is located symmetrically about the coordinate axes, this curve has the form shown in Fig. 47. The intersection points of the hyperbola with the focal axis are called its vertices. Assuming in the hyperbola equation, we find the abscissas of its vertices: . Thus, the hyperbola has two vertices: . The hyperbola does not intersect with the y-axis. In fact, putting in the hyperbola equation we get imaginary values for y: . Therefore, the focal axis of the hyperbola is called the real axis, and the symmetry axis perpendicular to the focal axis is called the imaginary axis of the hyperbola. The real axis is also called the segment connecting the vertices of the hyperbola, and its length is 2a. The segment connecting the points (see Fig. 47), as well as its length, is called the imaginary axis of the hyperbola. The numbers a and b are respectively called the real and imaginary semiaxes of the hyperbola. Consider now a hyperbola located in the first quadrant and which is the graph of the function Let us show that the points of this graph, located at a sufficiently large distance from the origin, are arbitrarily close to the straight line passing through the origin and having a slope To this end, consider two points that have the same abscissa and lie respectively on the curve (37) and the straight line (38) (Fig. 48), and make up the difference between the ordinates of these points The numerator of this fraction is a constant value, and the denominator increases indefinitely with an unlimited increase. Therefore, the difference tends to zero, i.e., the points M and N approach indefinitely with an unlimited increase in the abscissa. From the symmetry of the hyperbola with respect to the coordinate axes, it follows that there is another straight line , to which the points of the hyperbola are arbitrarily close at an unlimited distance from the origin. Direct are called asymptotes of the hyperbola. On fig. 49 shows the relative position of the hyperbola and its asymptotes. This figure also shows how to construct the asymptotes of the hyperbola. To do this, construct a rectangle centered at the origin and with sides parallel to the axes and, respectively, equal to . This rectangle is called the main rectangle. Each of its diagonals, extended indefinitely in both directions, is an asymptote of a hyperbola. Before constructing a hyperbola, it is recommended to build its asymptotes. The ratio of half the distance between the foci to the real semiaxis of the hyperbola is called the eccentricity of the hyperbola and is usually denoted by the letter: Since for a hyperbola, then the eccentricity of the hyperbola is greater than one: The eccentricity characterizes the shape of the hyperbola Indeed, it follows from formula (35) that . This shows that the smaller the eccentricity of the hyperbola, the smaller the ratio - of its semiaxes. But the relation - determines the shape of the main rectangle of the hyperbola, and hence the shape of the hyperbola itself. The smaller the eccentricity of the hyperbola, the more extended its main rectangle (in the direction of the focal axis). Hyperbola and parabola Let's move on to the second part of the article. about second order lines, dedicated to two other common curves - hyperbole and parabola. If you came to this page from a search engine or have not yet had time to navigate the topic, then I recommend that you first study the first section of the lesson, in which we examined not only the main theoretical points, but also got acquainted with ellipse. For the rest of the readers, I propose to significantly replenish their school knowledge about parabola and hyperbola. Hyperbola and parabola - is it simple? … Don't wait =) Hyperbola and its canonical equation The general structure of the presentation of the material will resemble the previous paragraph. Let's start with the general concept of a hyperbola and the problem of its construction. The canonical equation of a hyperbola has the form , where are positive real numbers. Note that, unlike ellipse, the condition is not imposed here, that is, the value of "a" may be less than the value of "be". I must say, quite unexpectedly ... the equation of the "school" hyperbole does not even closely resemble the canonical record. But this riddle will still have to wait for us, but for now let's scratch the back of our head and remember what characteristic features the curve under consideration has? Let's spread it on the screen of our imagination function graph …. A hyperbola has two symmetrical branches. The hyperbole has two asymptotes. Good progress! Any hyperbole has these properties, and now we will look with genuine admiration at the neckline of this line: Example 4 Construct a hyperbola given by the equation Decision: at the first step, we bring this equation to the canonical form . Please remember the typical procedure. On the right, you need to get a “one”, so we divide both parts of the original equation by 20: Here you can reduce both fractions, but it is more optimal to make each of them three-story: And only after that to carry out the reduction: We select the squares in the denominators: Why is it better to carry out transformations in this way? After all, the fractions of the left side can be immediately reduced and get. The fact is that in the example under consideration, we were a little lucky: the number 20 is divisible by both 4 and 5. In the general case, such a number does not work. Consider, for example, the equation . Here, with divisibility, everything is sadder and without three-story fractions no longer needed: So, let's use the fruit of our labors - the canonical equation: How to build a hyperbole? There are two approaches to constructing a hyperbola - geometric and algebraic. It is advisable to adhere to the following algorithm, first the finished drawing, then the comments: 1) First of all, we find asymptotes. If the hyperbola is given by the canonical equation , then its asymptotes are straight 2) Now we find two vertices of a hyperbola, which are located on the x-axis at points 3) We are looking for additional points. Usually 2-3 is enough. In the canonical position, the hyperbola is symmetrical about the origin and both coordinate axes, so it is enough to perform calculations for the 1st coordinate quarter. The technique is exactly the same as for the construction ellipse. From the canonical equation on the draft, we express: It suggests finding points with abscissas: 4) Draw the asymptotes on the drawing A technical difficulty can arise with an irrational slope factor, but this is a completely surmountable problem. Line segment called real axis hyperbole, In our example: Definition of a hyperbola. Foci and eccentricity In a hyperbole, in the same way as in ellipse, there are two singular points , which are called tricks. I didn’t say it, but just in case, suddenly someone misunderstands: the center of symmetry and the focus points, of course, do not belong to the curves. The general concept of the definition is also similar: Hyperbole is the set of all points in the plane, absolute value the difference in distances to each of which from two given points is a constant value, numerically equal to the distance between the vertices of this hyperbola: . In this case, the distance between the foci exceeds the length of the real axis: . If the hyperbola is given by the canonical equation, then distance from the center of symmetry to each of the foci calculated by the formula: . For the studied hyperbola: Let's go over the definition. Denote by the distances from the foci to an arbitrary point of the hyperbola: First, mentally move the blue dot along the right branch of the hyperbola - wherever we are, module(absolute value) the difference between the lengths of the segments will be the same: If the point is "thrown" to the left branch, and moved there, then this value will remain unchanged. The sign of the modulus is needed for the reason that the difference in lengths can be either positive or negative. By the way, for any point on the right branch Moreover, in view of the obvious property of the modulus, it does not matter what to subtract from what. Let's make sure that in our example the modulus of this difference is really equal to the distance between the vertices. Mentally place a point on the right vertex of the hyperbola. Then: , which was to be checked.Parametric equation of a hyperbola
Hyperbola and its canonical equation
How to build a hyperbole?
From a practical point of view, drawing with a compass ... I would even say utopian, so it is much more profitable to bring simple calculations to the rescue again.
Parabola and its canonical equation
determines the upper arc of the parabola, the equation determines the lower arc.
definition of a parabola:
In our example: 
The definition of a parabola is even easier to understand than the definitions of an ellipse and a hyperbola. For any point of the parabola, the length of the segment (the distance from the focus to the point) is equal to the length of the perpendicular (the distance from the point to the directrix): Rotation and translation of a parabola
From a practical point of view, drawing with a compass ... I would even say utopian, so it is much more profitable to bring simple calculations to the rescue again.
. In our case: 
. This item is required! This is a fundamental feature of the drawing, and it would be a gross mistake if the branches of the hyperbola “crawl out” beyond their asymptotes.
. It is derived elementarily: if , then the canonical equation turns into , whence it follows that . The considered hyperbola has vertices 
The equation breaks down into two functions: 
- defines the upper arcs of the hyperbola (what we need); 
- defines the lower arcs of the hyperbola.
, vertices , additional and symmetrical points in other coordinate quarters. We carefully connect the corresponding points at each branch of the hyperbola:
its length - the distance between the vertices;
number 
called real semiaxis hyperbole;
number – imaginary axis.
, and, obviously, if the given hyperbola is rotated around the center of symmetry and/or moved, then these values will not change.
And, accordingly, focuses have coordinates 
.
(because the segment is shorter than the segment ). For any point of the left branch, the situation is exactly opposite and 
.
