Parabola lying on its side. Geometric meaning of the parameter in the parabola equation

A parabola is the locus of points for each of which the distance to some fixed point on the plane, called the focus, is equal to the distance to some fixed line, called the directrix (it is assumed that this line does not pass through the focus).

The focus of a parabola is usually denoted by the letter F, distance from focus to directrix-letter r. Size p called parameter parabolas. The image of the parabola is given in Fig. 61 (the reader will receive a comprehensive explanation of this drawing after reading the next few paragraphs).

Comment. In accordance with the n° 100 says that the parabola has eccentricity =1.

Let some parabola be given (at the same time, we assume that the parameter p). Let us introduce a Cartesian rectangular coordinate system on the plane, the axes of which will be positioned in a special way with respect to this parabola. Namely, we draw the abscissa axis through the focus perpendicular to the directrix and consider it directed from the directrix to the focus; Let's place the origin of coordinates in the middle between focus and headmistress (Fig. 61). Let us derive the equation of this parabola in this coordinate system.

Let's take an arbitrary point on the plane M and denote its coordinates by X And u. Let us further denote by r distance from point M to focus (r=FM), through r- distance from point M to the headmistress. Dot M will be on a (given) parabola if and only if

To obtain the required equation, you need to replace the variables in equality (1) r And A their expressions through the current coordinates x, y. Note that the focus F has coordinates ; taking this into account and applying formula (2) n° 18. we find:

(2)

Let us denote by Q base of a perpendicular dropped from a point M to the headmistress. Obviously, period Q has coordinates ; from here and from formula (2) n° 18 we get:

(3),

(when extracting the root, we took with our sign, since - the number is positive; this follows from the fact that the point M(x;y) should be on the side of the director where the focus is, i.e. there should be x > , whence Replacing in equality (1) g and d their expressions (2) and (3), we find:

(4)

This is the equation of the parabola in question in the designated coordinate system, since it is satisfied by the coordinates of the point M(x;y) if and only if the point M lies on this parabola.

Wanting to obtain the parabola equation in a simpler form, let us square both sides of equality (4); we get:

(5),

We derived equation (6) as a consequence of equation (4). It is easy to show that equation (4) can in turn be derived as a consequence of equation (6). In fact, equation (5) is derived from equation (6) in an obvious way (“in reverse”); further, from equation (5) we have.

Definition: A parabola is the geometric locus of points on a plane for which the distance to some fixed point F of this plane is equal to the distance to some fixed straight line. Point F is called the focus of the parabola, and the fixed line is called the directrix of the parabola.

To derive the equation, let's build:

WITH according to the definition:

Since 2 >=0, the parabola lies in the right half-plane. As x increases from 0 to infinity
. The parabola is symmetrical about Ox. The point of intersection of a parabola with its axis of symmetry is called the vertex of the parabola.

45. Second order curves and their classification. The main theorem about kvp.

There are 8 types of KVP:

1.ellipses

2.hyperboles

3.parabolas

Curves 1,2,3 are canonical sections. If we intersect the cone with a plane parallel to the axis of the cone, we obtain a hyperbola. If the plane is parallel to the generatrix, then it is a parabola. All planes do not pass through the vertex of the cone. If it is any other plane, then it is an ellipse.

4. pair of parallel lines y 2 +a 2 =0, a0

5. pair of intersecting lines y 2 -k 2 x 2 =0

6.one straight line y 2 =0

7.one point x 2 + y 2 =0

8. empty set - empty curve (curve without points) x 2 + y 2 +1=0 or x 2 + 1=0

Theorem (main theorem about KVP): Equation of the form

a 11 x 2 + 2 a 12 x y + a 22 y 2 + 2 a 1 x + 2a 2 y+a 0 = 0

can only represent a curve of one of these eight types.

Idea of ​​proof is to move to a coordinate system in which the KVP equation will take the simplest form, when the type of curve it represents becomes obvious. The theorem is proven by rotating the coordinate system through an angle at which the term with the product of coordinates disappears. And with the help of parallel transfer of the coordinate system in which either the term with the x variable or the term with the y variable disappears.

Transition to a new coordinate system: 1. Parallel transfer

2. Rotate

45. Second order surfaces and their classification. The main theorem about pvp. Surfaces of rotation.

P VP - a set of points whose rectangular coordinates satisfy the 2nd degree equation: (1)

It is assumed that at least one of the coefficients of squares or products is different from 0. The equation is invariant with respect to the choice of coordinate system.

Theorem Any plane intersects the PVP along the CVP, with the exception of a special case when the entire plane is in the section. (The PVP can be a plane or a pair of planes).

There are 15 types of PVP. Let us list them, indicating the equations by which they are specified in suitable coordinate systems. These equations are called canonical (the simplest). Construct geometric images corresponding to canonical equations using the method of parallel sections: Intersect the surface with coordinate planes and planes parallel to them. The result is sections and curves that give an idea of ​​the shape of the surface.

1. Ellipsoid.

If a=b=c then we get a sphere.

2. Hyperboloids.

1). Single-sheet hyperboloid:

Section of a single-sheet hyperboloid by coordinate planes: XOZ:
- hyperbole.

YOZ:
- hyperbole.

XOY plane:
- ellipse.

2). Two-sheet hyperboloid.

The origin is a point of symmetry.

Coordinate planes are planes of symmetry.

Plane z = h intersects a hyperboloid along an ellipse
, i.e. plane z = h begins to intersect the hyperboloid at | h |  c. Section of a hyperboloid by planes x = 0 And y = 0 - these are hyperboles.

The numbers a, b, c in equations (2), (3), (4) are called the semi-axes of ellipsoids and hyperboloids.

3. Paraboloids.

1). Elliptical paraboloid:

Plane section z = h There is
, Where
. From the equation it is clear that z  0 is an infinite bowl.

Intersection of planes y = h And x= h
- this is a parabola and in general

2). Hyperbolic paraboloid:

Obviously, the XOZ and YOZ planes are planes of symmetry, the z axis is the axis of the paraboloid. Intersection of a paraboloid with a plane z = h– hyperboles:
,
. Plane z=0 intersects a hyperbolic paraboloid along two axes
which are asymptotes.

4. Cone and cylinders of the second order.

1). A cone is a surface
. The cone is formed by straight lines passing through the origin 0 (0, 0, 0). The cross section of a cone is an ellipse with semi-axes
.

2). Second order cylinders.

This is an elliptical cylinder
.

Whatever line we take that intersects the ellipses and is parallel to the Oz axis satisfies this equation. By moving this straight line around the ellipse we obtain a surface.

G hyperbolic cylinder:

On the XOU plane it is a hyperbola. We move the straight line intersecting the hyperbola parallel to Oz along the hyperbola.

Parabolic cylinder:

N and the XOU plane is a parabola.

Cylindrical surfaces are formed by a straight line (generative) moving parallel to itself along a certain straight line (guide).

10. Pair of intersecting planes

11.Pair of parallel planes

12.
- straight

13. Straight line - a “cylinder” built on one point

14.One point

15.Empty set

The main theorem about PVP: Each PVP belongs to one of the 15 types discussed above. There are no other PVP.

Surfaces of rotation. Let the PDSC Oxyz be given and in the Oyz plane the line e defined by the equation F(y,z)=0 (1). Let's create an equation for the surface obtained by rotating this line around the Oz axis. Let's take a point M(y,z) on line e. When the plane Oyz rotates around Oz, point M will describe a circle. Let N(X,Y,Z) be an arbitrary point of this circle. It is clear that z=Z.

.

Substituting the found values ​​of z and y into equation (1) we obtain the correct equality:
those. the coordinates of point N satisfy the equation
. Thus, any point on the surface of rotation satisfies equation (2). It is not difficult to prove that if a point N(x 1 ,y 1 ,z 1) satisfies equation (2) then it belongs to the surface in question. Now we can say that equation (2) is the desired equation for the surface of revolution.

Definition 1

A parabola is a curve formed by a geometric set of points located at the same distance from a certain point $F$, called the focus and not lying either on this curve or on the straight line $d$.

That is, the ratio of the distances from an arbitrary point on a parabola to the focus and from the same point to the directrix is ​​always equal to one, this ratio is called eccentricity.

The term “eccentricity” is also used for hyperbolas and ellipses.

Basic terms from the canonical parabola equation

Point $F$ is called the focus of the parabola, and line $d$ is its directrix.

The axis of symmetry of a parabola is a line passing through the vertex of the parabola $O$ and its focus $F$, so that it forms a right angle with the directrix $d$.

The vertex of a parabola is the point from which the distance to the directrix is ​​minimal. This point divides the distance from the focus to the directrix in half.

What is the canonical equation of a parabola?

Definition 2

The canonical equation of a parabola is quite simple, easy to remember and has the following form:

$y^2 = 2px$, where the number $p$ must be greater than zero.

The number $p$ from the equation is called the "focal parameter".

This equation of a parabola, or rather this formula most often used in higher mathematics, is applicable in the case when the axis of the parabola coincides with the $OX$ axis, that is, the parabola is located as if on its side.

A parabola described by the equation $x^2 = 2py$ is a parabola whose axis coincides with the $OY$ axis; we are accustomed to such parabolas at school.

And the parabola, which has a minus in front of the second part of the equation ($y^2 = - 2px$), is rotated 180° with respect to the canonical parabola.

A parabola is a special case of a 2nd order curve; accordingly, in general, the equation for a parabola looks exactly the same as for all such curves and is suitable for all cases, and not only when the parabola is parallel to $OX$.

In this case, the discriminant calculated by the formula $B^2 – 4AC$ is equal to zero, and the equation itself looks like this: $Ax^2 + B \cdot x \cdot y + C\cdot y^2 + D\cdot x + E\ cdot y + F = 0$

Derivation by graphing the canonical equation for a parabola

Figure 1. Graph and derivation of the canonical parabola equation

From the definition given above in this article, we will compose an equation for a parabola with the apex located at the intersection of the coordinate axes.

Using the existing graph, we determine from it the $x$ and $y$ points $F$ from the definition of a parabolic curve given above, $x = \frac(p)(2)$ and $y = 0$.

First, let's create an equation for the straight line $d$ and write it down: $x = - \frac(p)(2)$.

For an arbitrary point M lying on our curve, according to definition, the following relation is valid:

$FM$ = $MM_d$ (1), where $M_d$ is the intersection point of the perpendicular drawn from the point $M$ with the directrix $d$.

X and Y for this point are equal to $\frac(p)(2)$ $y$ respectively.

Let us write equation (1) in coordinate form:

$\sqrt((x - \frac(p)(2))^2 + y^2 )= x + \frac(p)(2)$

Now, in order to get rid of the root, you need to square both sides of the equation:

$(x - \frac(p)(2))^2 + y^2 = x^2 +px^2 + \frac(p^2)(4)$

After simplification, we obtain the canonical equation of the parabola: $y^2 = px$.

Parabola described by a quadratic function

The equation that describes a parabola with its apex located anywhere on the graph and not necessarily coinciding with the intersection of the coordinate axes looks like this:

$y = ax^2 + bx + c$.

To calculate $x$ and $y$ for the vertex of such a parabola, you need to use the following formulas:

$x_A = - \frac(b)(2a)$

$y_A = - \frac(D)(4a)$, where $D = b^2 – 4ac$.

Example 1

An example of composing a classic parabola equation

Task. Knowing the location of the focal point, create the canonical equation of the parabola. The coordinates of the focal point $F$ are $(4; 0)$.

Since we are considering a parabola, the graph of which is given by the canonical equation, its vertex $O$ is located at the intersection of the x and y axes, therefore the distance from the focus to the vertex is equal to $\frac(1)(2)$ of the focal parameter $\frac(p )(2) = $4. By simple calculations we find that the focal parameter itself is $p = 8$.

After substituting the value of $p$ into the canonical form of the equation, our equation becomes $y^2 = 16x$.

How to write a parabola equation using an existing graph

Example 2

Figure 2. Canonical equation for a parabola, graph and example for solution

First, we need to select point $M$, which belongs to the graph of our function, and, omitting the perpendiculars from it on the axes $OX$ and $OY$, write down its x and y, in our case, point $M$ is $(2;2) $.

Now we need to substitute the $x$ and $y$ obtained for this point into the canonical equation of the parabola $y^2 = px$, we get:

$2^2 = 2 \cdot 2p$

Reducing, we get the following parabola equation $y^2 = 2 \cdot x$.

Let us introduce a rectangular coordinate system, where . Let the axis pass through the focus F parabola and perpendicular to the directrix, and the axis passes midway between the focus and the directrix. Let us denote by the distance between the focus and the directrix. Then the directrix equation.

The number is called the focal parameter of the parabola. Let be the current point of the parabola. Let be the focal radius of the point of the hyperbola. Let be the distance from the point to the directrix. Then( drawing 27.)

Drawing 27.

By definition of a parabola. Hence,

Let's square the equation and get:

(15)

where (15) is the canonical equation of a parabola that is symmetrical about the axis and passes through the origin.

Study of the properties of a parabola

1) Vertex of the parabola:

Equation (15) is satisfied by numbers and, therefore, the parabola passes through the origin.

2) Parabola symmetry:

Let it belong to the parabola, i.e. true equality. The point is symmetrical to the point relative to the axis, therefore, the parabola is symmetrical relative to the abscissa axis.

Parabola eccentricity:

Definition 4.2. The eccentricity of a parabola is a number equal to one.

Since by definition of a parabola.

4) Tangent of the parabola:

The tangent to a parabola at the point of tangency is given by the equation

Where ( drawing 28.)

Drawing 28.

Parabola image

Drawing 29.

Using ESO-Mathcad:

drawing 30.)

Drawing 30.

a) Construction without the use of ICT: To construct a parabola, we set a rectangular coordinate system with a center at point O and a unit segment. We mark the focus on the OX axis, since we draw such that, and the directrix of the parabola. We construct a circle at a point with a radius equal to the distance from the straight line to the directrix of the parabola. The circle intersects the line at points . We construct a parabola so that it passes through the origin and through the points.( drawing 31.)

Drawing 31.

b)Using ESO-Mathcad:

The resulting equation looks like: . To construct a second-order line in the Mathcad program, we reduce the equation to the form: .( drawing 32.)

Drawing 32.

In order to summarize the work on the theory of second-order lines in elementary mathematics and for the convenience of using information about lines when solving problems, we will include all the data on second-order lines in Table No. 1.

Table No. 1.

Second order lines in elementary mathematics

Possibilities of using ICT in the study of second-order lines

The process of informatization, which today has covered all aspects of the life of modern society, has several priority areas, which, of course, should include the informatization of education. It is the fundamental basis for the global rationalization of human intellectual activity through the use of information and communication technologies (ICT).

The mid-90s of the last century until today is characterized by the widespread use and availability of personal computers in Russia, the widespread use of telecommunications, which allows the introduction of developed educational information technologies into the educational process, improving and modernizing it, improving the quality of knowledge, increasing motivation to learn, making maximum use of the principle of individualization of learning. Information technologies for education are a necessary tool at this stage of informatization of education.

Information technologies not only facilitate access to information and open up opportunities for variability in educational activities, their individualization and differentiation, but also make it possible to reorganize the interaction of all subjects of learning in a new way, to build an educational system in which the student would be an active and equal participant in educational activities.

The formation of new information technologies within the framework of subject lessons stimulates the need to create new software and methodological complexes aimed at qualitatively increasing the effectiveness of the lesson. Therefore, for the successful and purposeful use of information technology tools in the educational process, teachers must know the general description of the principles of operation and the didactic capabilities of software applications, and then, based on their experience and recommendations, “build” them into the educational process.

The study of mathematics is currently associated with a number of features and difficulties in the development of school education in our country.

A so-called crisis in mathematics education has emerged. The reasons for this are as follows:

In changing priorities in society and in science, that is, the priority of the humanities is currently growing;

In reducing the number of mathematics lessons in school;

The isolation of the content of mathematical education from life;

Has little impact on the feelings and emotions of students.

Today the question remains open: “How to most effectively use the potential capabilities of modern information and communication technologies when teaching schoolchildren, including when teaching mathematics?”

A computer is an excellent assistant in studying a topic such as “Quadratic Function”, because using special programs you can build graphs of various functions, explore the function, easily determine the coordinates of intersection points, calculate the areas of closed figures, etc. For example, in a 9th grade algebra lesson devoted to graph transformation (stretching, compressing, moving coordinate axes), you can only see the frozen result of the construction, while the entire dynamics of the sequential actions of the teacher and student can be seen on the monitor screen.

The computer, like no other technical tool, accurately, visually and excitingly reveals ideal mathematical models to the student, i.e. what a child should strive for in his practical actions.

How many difficulties does a mathematics teacher have to go through in order to convince students that the tangent to the graph of a quadratic function at the point of tangency practically merges with the graph of the function. It is very easy to demonstrate this fact on a computer—it is enough to narrow the interval along the Ox axis and discover that in a very small neighborhood of the tangency point, the graph of the function and the tangent line coincide. All these actions take place in front of the students. This example provides an impetus for active reflection in the lesson. The use of a computer is possible both during the explanation of new material in class and at the control stage. With the help of these programs, for example “My Test”, the student can independently test his level of knowledge in theory and complete theoretical and practical tasks. The programs are convenient due to their versatility. They can be used for both self-control and teacher control.

Reasonable integration of mathematics and computer technology will allow us to take a richer and deeper look at the process of solving a problem and the process of understanding mathematical laws. In addition, the computer will help to form a graphic, mathematical and mental culture of students, and with the help of a computer you can prepare didactic materials: cards, survey sheets, tests, etc. At the same time, give the children the opportunity to independently develop tests on the topic, during which interest and creative approach.

Thus, there is a need to use computers in mathematics lessons as widely as possible. The use of information technology will help improve the quality of knowledge, expand the horizons of studying the quadratic function, and therefore help to find new prospects for maintaining students’ interest in the subject and topic, and therefore for a better, more attentive attitude towards it. Today, modern information technologies are becoming the most important tool for modernizing the school as a whole - from management to education and ensuring accessibility of education.

A parabola is a set of points in a plane equidistant from a given point(focus)and from a given line not passing through a given point (headmistresses), located in the same plane(Fig. 5).

In this case, the coordinate system is chosen so that the axis
passes perpendicular to the directrix through the focus, its positive direction is chosen from the directrix towards the focus. The ordinate axis runs parallel to the directrix, midway between the directrix and the focus, whence the directrix equation
, focus coordinates
. The origin is the vertex of the parabola, and the x-axis is its axis of symmetry. Parabola eccentricity
.

In a number of cases, parabolas defined by the equations are considered

A)

b)
(for all cases
)

V)
.

In case a) the parabola is symmetrical about the axis
and is directed in its negative direction (Fig. 6).

In cases b) and c) the axis of symmetry is the axis
(Fig. 6). Focus coordinates for these cases:

A)
b)
V)
.

Directrix equation:

A)
b)
V)
.

Example 4. A parabola with a vertex at the origin passes through a point
and symmetrical about the axis
. Write its equation.

Solution:

Since the parabola is symmetrical about the axis
and passes through the point with a positive abscissa, then it has the form shown in Fig. 5.

Substituting point coordinates into the equation of such a parabola
, we get
, i.e.
.

Therefore, the required equation

,

the focus of this parabola
, directrix equation
.

4. Transformation of the second order line equation to canonical form.

The general equation of the second degree has the form

where are the coefficients
do not go to zero at the same time.

Any line defined by equation (6) is called a second-order line. Using a transformation of the coordinate system, the equation of a second-order line can be reduced to its simplest (canonical) form.

1. In equation (6)
. In this case, equation (6) has the form

It is converted to its simplest form using parallel translation of coordinate axes according to the formulas

(8)

Where
– coordinates of the new beginning
(in the old coordinate system). New axles
And
parallel to the old ones. Dot
is the center of an ellipse or hyperbola and the vertex in the case of a parabola.

It is convenient to reduce equation (7) to its simplest form using the method of isolating complete squares, similar to how it was done for a circle.

Example 5. Reduce the second order line equation to its simplest form. Determine the type and location of this line. Find the coordinates of the foci. Make a drawing.

Solution:

We group members containing only and only , taking out the coefficients for And behind the bracket:

We complete the expressions in brackets to complete squares:

Thus, this equation is transformed to the form

We designate

or

Comparing with equations (8), we see that these formulas determine the parallel transfer of coordinate axes to the point
. In the new coordinate system, the equation will be written as follows:

Moving the free term to the right and dividing by it, we get:

.

So, this second-order line is an ellipse with semi-axes
,
. The center of the ellipse is at the new origin
, and its focal axis is the axis
. Distance of focuses from center, so new coordinates of right focus
. The old coordinates of the same focus are found from the parallel translation formulas:

Likewise, the new left focus coordinates
,
. His old coordinates:
,
.

Comment. To clarify the drawing, it is useful to find the intersection points of this line (7) with the old coordinate axes. To do this, we must first put in formula (7)
and then
and solve the resulting equations.

The appearance of complex roots will mean that line (7) does not intersect the corresponding coordinate axis.

For example, for the ellipse of the problem just discussed, the following equations are obtained:

The second of these equations has complex roots, so the ellipse axis
does not cross. The roots of the first equation are:

At points
And
ellipse intersects axis
(Fig. 7).

Example 6. Reduce the equation of a second order line to its simplest form. Determine the type and location of the line, find the focal coordinates.

Solution:

Since the member with missing, then you need to select a complete square only by :

We also take out the coefficient for

.

We designate

or

This results in a parallel transfer of the coordinate system to the point
. After translation, the equation will take the form

.

Therefore the focus has new coordinates

.

His old coordinates

If we put in this equation
or
, then we find that the parabola intersects the axis
at the point
, and the axis
she doesn't cross.

2. In equation (1)
. General equation (1) of the second degree is transformed to form (2), i.e. to that discussed in paragraph 1. case, by rotating the coordinate axes by an angle
according to formulas

(9)

Where
– new coordinates. Corner
is found from the equation

The coordinate axes are rotated so that the new axes
And
were parallel to the symmetry axes of the second order line.

Knowing
, can be found
And
using trigonometry formulas

,
.

If the rotation angle
agree to be considered acute, then in these formulas we must take the plus sign, and for
we must also take a positive solution to equation (5).

In particular, when
the coordinate system must be rotated by an angle
. The rotation formulas for coals look like:

(11)

Example 7. Reduce the second order line equation to its simplest form. Set the type and location of this line.

Solution:

In this case
, 1
,
, so the rotation angle
is found from the equation

.

Solution to this equation
And
. Limited to an acute angle
, let's take the first of them. Then

,

,
.

Substituting these values And into this equation

Opening the brackets and bringing similar ones, we get

.

Finally, dividing by the dummy term, we arrive at the equation of the ellipse

.

It follows that
,
, and the major axis of the ellipse is directed along the axis
, and the small one – along the axis
.

You get a point
, whose radius
inclined to the axis
at an angle
, for which
. Therefore, through this point
and a new x-axis will pass. Then we mark on the axes
And
the vertices of the ellipse and draw an ellipse (Fig. 9).

Note that this ellipse intersects the old coordinate axes at points that are found from quadratic equations (if we put in this equation
or
):

And
.