Throughout this chapter it is assumed that a certain scale has been chosen in the plane (in which all the figures considered below lie); Only rectangular coordinate systems with this scale are considered.
§ 1. Parabola
A parabola is known to the reader from a school mathematics course as a curve, which is the graph of a function
(Fig. 76). (1)
Graph of any quadratic trinomial
is also a parabola; is possible by simply shifting the coordinate system (by some vector OO), i.e. transforming
ensure that the graph of the function (in the second coordinate system) coincides with graph (2) (in the first coordinate system).
In fact, let us substitute (3) into equality (2). We get
We want to choose so that the coefficient at and the free term of the polynomial (with respect to ) on the right side of this equality are equal to zero. To do this, we determine from the equation
which gives
Now we determine from the condition
into which we substitute the already found value. We get
So, by means of shift (3), in which
we moved to a new coordinate system, in which the equation of the parabola (2) took the form
(Fig. 77).
Let's return to equation (1). It can serve as the definition of a parabola. Let us recall its simplest properties. A curve has an axis of symmetry: if a point satisfies equation (1), then the point symmetrical point M relative to the ordinate axis also satisfies equation (1) - the curve is symmetrical relative to the ordinate axis (Fig. 76).
If , then parabola (1) lies in the upper half-plane, having a single common point O with the abscissa axis.
With an unlimited increase in the absolute value of the abscissa, the ordinate also increases without limit. A general view of the curve is shown in Fig. 76, a.
If (Fig. 76, b), then the curve is located in the lower half-plane symmetrically relative to the abscissa axis to the curve.
If we move to a new coordinate system obtained from old replacement positive direction of the ordinate axis to the opposite, then the parabola, which has the equation y in the old system, will receive the equation y in the new coordinate system. Therefore, when studying parabolas, we can limit ourselves to equations (1), in which .
Let us finally change the names of the axes, i.e., we will move to a new coordinate system, in which the ordinate axis will be the old abscissa axis, and the abscissa axis will be the old ordinate axis. In this new system, equation (1) will be written in the form
Or, if the number is denoted by , in the form
Equation (4) is called in analytical geometry the canonical equation of a parabola; the rectangular coordinate system in which a given parabola has equation (4) is called the canonical coordinate system (for this parabola).
Now we will establish the geometric meaning of the coefficient. To do this we take the point
called the focus of parabola (4), and the straight line d, defined by the equation
This line is called the directrix of the parabola (4) (see Fig. 78).
Let be an arbitrary point of the parabola (4). From equation (4) it follows that Therefore, the distance of the point M from the directrix d is the number
The distance of point M from focus F is
But, therefore
So, all points M of the parabola are equidistant from its focus and directrix:
Conversely, every point M that satisfies condition (8) lies on the parabola (4).
Indeed,
Hence,
and, after opening the parentheses and bringing like terms,
We have proven that each parabola (4) is the locus of points equidistant from the focus F and from the directrix d of this parabola.
At the same time, we have established the geometric meaning of the coefficient in equation (4): the number is equal to the distance between the focus and the directrix of the parabola.
Let us now assume that a point F and a line d not passing through this point are given arbitrarily on the plane. Let us prove that there exists a parabola with focus F and directrix d.
To do this, draw a line g through point F (Fig. 79), perpendicular to line d; let us denote the point of intersection of both lines by D; the distance (i.e. the distance between point F and straight line d) will be denoted by .
Let us turn the straight line g into an axis, taking the direction DF on it as positive. Let us make this axis the abscissa axis of a rectangular coordinate system, the origin of which is the middle O of the segment
Then straight line d also receives the equation .
Now we can write the canonical equation of the parabola in the selected coordinate system:
where point F will be the focus, and straight line d will be the directrix of the parabola (4).
We established above that a parabola is the locus of points M equidistant from point F and line d. So, we can give such a geometric (i.e., independent of any coordinate system) definition of a parabola.
Definition. A parabola is the locus of points equidistant from some fixed point (the “focus” of the parabola) and some fixed line (the “directrix” of the parabola).
Denoting the distance between the focus and the directrix of a parabola by , we can always find a rectangular coordinate system that is canonical for a given parabola, that is, one in which the equation of the parabola has the canonical form:
Conversely, any curve that has such an equation in some rectangular coordinate system is a parabola (in the geometric sense just established).
The distance between the focus and the directrix of a parabola is called the focal parameter, or simply the parameter of the parabola.
The line passing through the focus perpendicular to the directrix of the parabola is called its focal axis (or simply axis); it is the axis of symmetry of the parabola - this follows from the fact that the axis of the parabola is the abscissa axis in the coordinate system, relative to which the equation of the parabola has the form (4).
If a point satisfies equation (4), then a point symmetrical to the point M relative to the abscissa axis also satisfies this equation.
The point of intersection of a parabola with its axis is called the vertex of the parabola; it is the origin of the coordinate system canonical for a given parabola.
Let's give another geometric interpretation of the parabola parameter.
Let us draw a straight line through the focus of the parabola, perpendicular to the axis of the parabola; it will intersect the parabola at two points (see Fig. 79) and determine the so-called focal chord of the parabola (i.e., the chord passing through the focus parallel to the directrix of the parabola). Half the length of the focal chord is the parameter of the parabola.
In fact, half the length of the focal chord is the absolute value of the ordinate of any of the points, the abscissa of each of which is equal to the abscissa of the focus, i.e. Therefore, for the ordinate of each point we have
Q.E.D.
Consider a line on the plane and a point not lying on this line. AND ellipse, And hyperbola can be defined in a unified way as the geometric locus of points for which the ratio of the distance to a given point to the distance to a given straight line is a constant value
rank ε. At 0 1 - hyperbola. The parameter ε is eccentricity of both ellipse and hyperbola. Of the possible positive values one parameter ε, namely ε = 1, turns out to be unused. This value corresponds to the geometric locus of points equidistant from a given point and from a given line.
Definition 8.1. The locus of points in a plane equidistant from a fixed point and from a fixed line is called parabola.
The fixed point is called focus of the parabola, and the straight line - directrix of a parabola. At the same time, it is believed that parabola eccentricity equal to one.
From geometric considerations it follows that the parabola is symmetrical with respect to the straight line perpendicular to the directrix and passing through the focus of the parabola. This straight line is called the axis of symmetry of the parabola or simply parabola axis. A parabola intersects its axis of symmetry at a single point. This point is called the vertex of the parabola. It is located in the middle of the segment connecting the focus of the parabola with the point of intersection of its axis with the directrix (Fig. 8.3).
Parabola equation. To derive the equation of a parabola, we choose on the plane origin at the vertex of the parabola, as x-axis- the axis of the parabola, the positive direction on which is specified by the position of the focus (see Fig. 8.3). This coordinate system is called canonical for the parabola in question, and the corresponding variables are canonical.
Let us denote the distance from the focus to the directrix by p. He is called focal parameter of the parabola.
Then the focus has coordinates F(p/2; 0), and the directrix d is described by the equation x = - p/2. The locus of points M(x; y), equidistant from the point F and from the line d, is given by the equation
Let us square equation (8.2) and present similar ones. We get the equation
which is called canonical parabola equation.
Note that squaring in this case is an equivalent transformation of equation (8.2), since both sides of the equation are non-negative, as is the expression under the radical.
Type of parabola. If the parabola y 2 = x, the form of which we consider known, is compressed with a coefficient of 1/(2p) along the x-axis, then we get a parabola general view, which is described by equation (8.3).
Example 8.2. Let us find the coordinates of the focus and the equation of the directrix of a parabola if it passes through a point whose canonical coordinates are (25; 10).
In canonical coordinates, the equation of the parabola has the form y 2 = 2px. Since the point (25; 10) is on the parabola, then 100 = 50p and therefore p = 2. Therefore, y 2 = 4x is the canonical equation of the parabola, x = - 1 is the equation of its directrix, and the focus is at the point (1; 0 ).
Optical property of a parabola. The parabola has the following optical property. If a light source is placed at the focus of the parabola, then all light rays after reflection from the parabola will be parallel to the axis of the parabola (Fig. 8.4). The optical property means that at any point M of the parabola normal vector the tangent makes equal angles with the focal radius MF and the abscissa axis.
A function of the form where is called quadratic function.
Graph of a quadratic function – parabola.
Let's consider the cases:
I CASE, CLASSICAL PARABOLA
That is , ,
To construct, fill out the table by substituting the x values into the formula:
Mark the points (0;0); (1;1); (-1;1), etc. on the coordinate plane (the smaller the step we take the x values (in this case, step 1), and the more x values we take, the smoother the curve will be), we get a parabola:
It is easy to see that if we take the case , , , that is, then we get a parabola that is symmetrical about the axis (oh). It’s easy to verify this by filling out a similar table:
II CASE, “a” IS DIFFERENT FROM UNIT
What will happen if we take , , ? How will the behavior of the parabola change? With title="Rendered by QuickLaTeX.com" height="20" width="55" style="vertical-align: -5px;"> парабола изменит форму, она “похудеет” по сравнению с параболой (не верите – заполните соответствующую таблицу – и убедитесь сами):!}
In the first picture (see above) it is clearly visible that the points from the table for the parabola (1;1), (-1;1) were transformed into points (1;4), (1;-4), that is, with the same values, the ordinate of each point is multiplied by 4. This will happen to all key points of the original table. We reason similarly in the cases of pictures 2 and 3.
And when the parabola “becomes wider” than the parabola:
Let's summarize:
1)The sign of the coefficient determines the direction of the branches. With title="Rendered by QuickLaTeX.com" height="14" width="47" style="vertical-align: 0px;"> ветви направлены вверх, при - вниз. !}
2) Absolute value coefficient (modulus) is responsible for the “expansion” and “compression” of the parabola. The larger , the narrower the parabola; the smaller |a|, the wider the parabola.
III CASE, “C” APPEARS
Now let's introduce into the game (that is, consider the case when), we will consider parabolas of the form . It is not difficult to guess (you can always refer to the table) that the parabola will shift up or down along the axis depending on the sign:
IV CASE, “b” APPEARS
When will the parabola “break away” from the axis and finally “walk” along the entire coordinate plane? When will it stop being equal?
Here to construct a parabola we need formula for calculating the vertex: , .
So at this point (as at point (0;0) new system coordinates) we will build a parabola, which we can already do. If we are dealing with the case, then from the vertex we put one unit segment to the right, one up, - the resulting point is ours (similarly, a step to the left, a step up is our point); if we are dealing with, for example, then from the vertex we put one unit segment to the right, two - upward, etc.
For example, the vertex of a parabola:
Now the main thing to understand is that at this vertex we will build a parabola according to the parabola pattern, because in our case.
When constructing a parabola after finding the coordinates of the vertex veryIt is convenient to consider the following points:
1) parabola will definitely pass through the point . Indeed, substituting x=0 into the formula, we obtain that . That is, the ordinate of the point of intersection of the parabola with the axis (oy) is . In our example (above), the parabola intersects the ordinate at point , since .
2) axis of symmetry parabolas is a straight line, so all points of the parabola will be symmetrical about it. In our example, we immediately take the point (0; -2) and build it symmetrical relative to the symmetry axis of the parabola, we get the point (4; -2) through which the parabola will pass.
3) Equating to , we find out the points of intersection of the parabola with the axis (oh). To do this, we solve the equation. Depending on the discriminant, we will get one (, ), two ( title="Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">, ) или нИсколько () точек пересечения с осью (ох) !} . In the previous example, our root of the discriminant is not an integer; when constructing, it doesn’t make much sense for us to find the roots, but we clearly see that we will have two points of intersection with the axis (oh) (since title="Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">), хотя, в общем, это видно и без дискриминанта.!}
So let's work it out
Algorithm for constructing a parabola if it is given in the form
1) determine the direction of the branches (a>0 – up, a<0 – вниз)
2) we find the coordinates of the vertex of the parabola using the formula , .
3) we find the point of intersection of the parabola with the axis (oy) using the free term, construct a point symmetrical to this point with respect to the symmetry axis of the parabola (it should be noted that it happens that it is unprofitable to mark this point, for example, because the value is large... we skip this point...)
4) At the found point - the vertex of the parabola (as at the point (0;0) of the new coordinate system) we construct a parabola. If title="Rendered by QuickLaTeX.com" height="20" width="55" style="vertical-align: -5px;">, то парабола становится у’же по сравнению с , если , то парабола расширяется по сравнению с !}
5) We find the points of intersection of the parabola with the axis (oy) (if they have not yet “surfaced”) by solving the equation
Example 1
Example 2
Note 1. If the parabola is initially given to us in the form , where are some numbers (for example, ), then it will be even easier to construct it, because we have already been given the coordinates of the vertex . Why?
Let's take a quadratic trinomial and isolate the complete square in it: Look, we got that , . You and I previously called the vertex of a parabola, that is, now,.
For example, . We mark the vertex of the parabola on the plane, we understand that the branches are directed downward, the parabola is expanded (relative to ). That is, we carry out points 1; 3; 4; 5 from the algorithm for constructing a parabola (see above).
Note 2. If the parabola is given in a form similar to this (that is, presented as a product of two linear factors), then we immediately see the points of intersection of the parabola with the axis (ox). In this case – (0;0) and (4;0). For the rest, we act according to the algorithm, opening the brackets.
Level III
3.1. Hyperbole touches lines 5 x – 6y – 16 = 0, 13x – 10y– – 48 = 0. Write down the equation of the hyperbola provided that its axes coincide with the coordinate axes.
3.2. Write equations for tangents to a hyperbola
1) passing through a point A(4, 1), B(5, 2) and C(5, 6);
2) parallel to straight line 10 x – 3y + 9 = 0;
3) perpendicular to straight line 10 x – 3y + 9 = 0.
Parabola is the geometric locus of points in the plane whose coordinates satisfy the equation
Parabola parameters:
Dot F(p/2, 0) is called focus parabolas, magnitude p – parameter , dot ABOUT(0, 0) – top . In this case, the straight line OF, about which the parabola is symmetrical, defines the axis of this curve.
 |
Magnitude 
Where M(x, y) – an arbitrary point of a parabola, called focal radius
, straight D: x = –p/2 – headmistress
(it does not intersect the interior region of the parabola). Magnitude 
is called the eccentricity of the parabola.
The main characteristic property of a parabola: all points of the parabola are equidistant from the directrix and focus (Fig. 24).
There are other forms of the canonical parabola equation that determine other directions of its branches in the coordinate system (Fig. 25):
 |
For parametric definition of a parabola as a parameter t the ordinate value of the parabola point can be taken:
Where t– arbitrary real number.
Example 1. Determine the parameters and shape of a parabola using its canonical equation:
Solution. 1. Equation y 2 = –8x defines a parabola with vertex at point ABOUT Oh. Its branches are directed to the left. Comparing this equation with the equation y 2 = –2px, we find: 2 p = 8, p = 4, p/2 = 2. Therefore, the focus is at the point F(–2; 0), directrix equation D: x= 2 (Fig. 26).
 |
2. Equation x 2 = –4y defines a parabola with vertex at point O(0; 0), symmetrical about the axis Oy. Its branches are directed downwards. Comparing this equation with the equation x 2 = –2py, we find: 2 p = 4, p = 2, p/2 = 1. Therefore, the focus is at the point F(0; –1), directrix equation D: y= 1 (Fig. 27).
Example 2. Determine parameters and type of curve x 2 + 8x – 16y– 32 = 0. Make a drawing.
Solution. Let's transform the left side of the equation using the complete square extraction method:
x 2 + 8x– 16y – 32 =0;
(x + 4) 2 – 16 – 16y – 32 =0;
(x + 4) 2 – 16y – 48 =0;
(x + 4) 2 – 16(y + 3).
As a result we get
(x + 4) 2 = 16(y + 3).
This is the canonical equation of a parabola with the vertex at the point (–4, –3), the parameter p= 8, branches pointing upward (), axis x= –4. Focus is on point F(–4; –3 + p/2), i.e. F(–4; 1) Headmistress D given by the equation y = –3 – p/2 or y= –7 (Fig. 28).
Example 4. Write an equation for a parabola with its vertex at the point V(3; –2) and focus at the point F(1; –2).
Solution. The vertex and focus of a given parabola lie on a straight line parallel to the axis Ox(same ordinates), the branches of the parabola are directed to the left (the abscissa of the focus is less than the abscissa of the vertex), the distance from the focus to the vertex is p/2 = 3 – 1 = 2, p= 4. Hence, the required equation
(y+ 2) 2 = –2 4( x– 3) or ( y + 2) 2 = = –8(x – 3).
Tasks for independent decision
I level
1.1. Determine the parameters of the parabola and construct it:
1) y 2 = 2x; 2) y 2 = –3x;
3) x 2 = 6y; 4) x 2 = –y.
1.2. Write the equation of a parabola with its vertex at the origin if you know that:
1) the parabola is located in the left half-plane symmetrically relative to the axis Ox And p = 4;
2) the parabola is located symmetrically relative to the axis Oy and passes through the point M(4; –2).
3) the directrix is given by equation 3 y + 4 = 0.
1.3. Write an equation for a curve all points of which are equidistant from the point (2; 0) and the straight line x = –2.
Level II
2.1. Determine the type and parameters of the curve.
I suggest that the rest of the readers significantly expand their school knowledge about parabolas and hyperbolas. Hyperbola and parabola - are they simple? ...Can't wait =)
Hyperbola and its canonical equation
General structure the presentation of the material will resemble the previous paragraph. Let's start with general concept hyperbolas and problems for its construction.
The canonical equation of a hyperbola has the form , where are positive real numbers. Please 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” hyperbola does not even closely resemble the canonical notation. But this mystery will still have to wait for us, but for now let’s scratch our heads and remember what characteristic features does the curve in question have? Let's spread it on the screen of our imagination graph of a function ….
A hyperbola has two symmetrical branches.
Not bad progress! Any hyperbole has these properties, and now we will look with genuine admiration at the neckline of this line:
Example 4
Construct the hyperbola given by the equation
Solution: in the first step, we bring this equation to canonical form. Please remember the standard procedure. On the right you need to get “one”, so we divide both sides of the original equation by 20: 
Here you can reduce both fractions, but it is more optimal to do each of them three-story:
And only after that carry out the reduction:
Select the squares in the denominators:
Why is it better to carry out transformations this way? After all, the fractions on the left side can be immediately reduced and obtained. 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 possible: 
So, let's use the fruit of our labors - the canonical equation:
How to construct a hyperbola?
There are two approaches to constructing a hyperbola - geometric and algebraic.
From a practical point of view, drawing with a compass... I would even say utopian, so it is much more profitable to once again use simple calculations to help.
It is advisable to adhere to the following algorithm, first finished drawing, then comments:
In practice, a combination of rotation by an arbitrary angle and parallel translation of the hyperbola is often encountered. This situation discussed in class Reducing the 2nd order line equation to canonical form.
Parabola and its canonical equation
It's finished! She's the one. Ready to reveal many secrets. The canonical equation of a parabola has the form , where is a real number. It is easy to notice that in its standard position the parabola “lies on its side” and its vertex is at the origin. In this case, the function specifies the upper branch of this line, and the function – the lower branch. It is obvious that the parabola is symmetrical about the axis. Actually, why bother:
Example 6
Construct a parabola
Solution: the top is known, let's find it additional points. The equation 
determines the upper arc of the parabola, the equation determines the lower arc.
In order to shorten the recording of the calculations, we will carry out the calculations “with one brush”: 
For compact recording, the results could be summarized in a table.
Before performing an elementary point-by-point drawing, let’s formulate a strict
definition of parabola:
A parabola is the set of all points in the 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 (spelled 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. In this case . In this case, the focus has coordinates , and the directrix is given by the equation .
In our example: 
The definition of a parabola is even simpler to understand than the definitions of an ellipse and a hyperbola. For any point on a 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):
Congratulations! Many of you have made a real discovery today. It turns out that a hyperbola and a parabola are not graphs of “ordinary” functions at all, but have a pronounced geometric origin.
Obviously, with an increase in the focal parameter, the branches of the graph will “raise” up and down, approaching infinitely close to the axis. As the “pe” value decreases, they will begin to compress and stretch along the axis
The eccentricity of any parabola is equal to unity:
Rotation and parallel translation of a parabola
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 discuss typical options for the location of this curve.
! Note : as in the cases with previous curves, it is more correct to talk about rotation and parallel translation coordinate axes, but the author will limit himself to a simplified version of the presentation so that the reader has a basic understanding of these transformations.
