Lesson “The volume of a cone. Study of the theory of conic sections Given a straight circular cone

Introduction

Relevance of the research topic.Conical sections were already known to mathematicians Ancient Greece (for example, Menekhmu, 4th century BC); with the help of these curves, some construction tasks (doubling a cube, etc.) were solved, which turned out to be inaccessible when using the simplest drawing tools - a compass and a ruler. In the first studies that have come down to us, Greek geometers obtained conical sections by drawing a cutting plane perpendicular to one of the generatrices, while, depending on the opening angle at the apex of the cone (i.e., the largest angle between the generatrices of one cavity), the intersection line turned out to be an ellipse, if this angle is acute, a parabola, if it is a right angle, and hyperbola, if it is obtuse. The most complete work on these curves was the "Conical Sections" by Apollonius of Perga (about 200 BC). Further advances in the theory of conic sections are associated with the creation in the 17th century. new geometric methods: projective (French mathematicians J. Desargues, B. Pascal) and, in particular, coordinate (French mathematicians R. Descartes, P. Fermat).

Interest in tapered sections has always been supported by the fact that these curves are often found in various phenomena nature and in human activity. In science, conical sections acquired special significance after the German astronomer I. Kepler discovered from observations, and the English scientist I. Newton theoretically substantiated the laws of planetary motion, one of which claims that planets and comets Solar system move along conical sections, in one of the focuses of which is the sun. The following examples refer to individual types of conic sections: a parabola describes a projectile or a stone thrown obliquely towards the horizon ( correct shape the curve is somewhat distorted by air resistance); some mechanisms use elliptical cogwheels ("elliptical cogwheels"); hyperbole serves as a graph inverse proportion, which is often observed in nature (for example, Boyle's law - Mariotte).

Objective:

Study of the theory of conic sections.

Research topic:

Conical sections.

Purpose of the study:

Theoretically study the features of conic sections.

Object of study:

Conical sections.

Subject of study:

Historical development of conic sections.

1. Formation of conic sections and their types

Tapered sections are lines that are formed in the section of a straight circular cone with different planes.

Note that a conical surface is a surface formed by the motion of a straight line passing all the time through a fixed point (the top of the cone) and intersecting all the time a fixed curve - a guide (in our case, a circle).

By classifying these lines by the nature of the location of the secant planes relative to the generatrices of the cone, curves of three types are obtained:

I. Curves formed by the section of a cone by planes not parallel to any of the generators. These curves will be various circles and ellipses. These curves are called elliptical curves.

II. Curves formed by the section of the cone by planes, each of which is parallel to one of the generatrices of the cone (Fig. 1 b). Only parabolas will be such curves.

III. Curves formed by the section of the cone by planes, each of which is parallel to some two generators (Fig. 1 c). such curves are hyperboles.

There can no longer be any IV type of curves, since there can be no plane parallel to three generators of the cone at once, since no three generators of the cone themselves already lie in the same plane.

Note that the cone can be crossed by planes in such a way that two straight lines are obtained in the section. For this, the secant planes must be drawn through the apex of the cone.

2. Ellipse

Two theorems are important for studying the properties of conic sections:

Theorem 1. Let a straight circular cone be given, which is cut by planes b 1, b 2, b 3, perpendicular to its axis. Then all the segments of the generatrices of the cone between any pair of circles (obtained in the section with the given planes) are equal to each other, i.e. A 1 B 1 \u003d A 2 B 2 \u003d etc. and B 1 C 1 \u003d B 2 C 2 \u003d etc. Theorem 2. If a spherical surface is given and some point S is outside it, then the segments of the tangents drawn from the point S to the spherical surface will be equal to each other, that is, SA 1 \u003d SA 2 \u003d SA 3, etc.

2.1 The main property of an ellipse

We dissect a straight circular cone with a plane intersecting all its generators. In section we get an ellipse. Draw a plane through the axis of the cone, perpendicular to the plane.

Let us inscribe two balls into the cone so that, being located on opposite sides of the plane and touching the conical surface, each of them touches the plane at some point.

Let one ball touch the plane at point F 1 and touch the cone along the circle С 1, and the other - at the point F 2 and touch the cone around the circle С 2.

Take an arbitrary point P on the ellipse.

This means that all conclusions drawn about it will be valid for any point of the ellipse. Draw the generator of the OP of the cone and mark the points R 1 and R 2 at which it touches the constructed balls.

Let's connect point P with points F 1 and F 2. Then РF 1 \u003d РR 1 and РF 2 \u003d РR 2, since РF 1, РR 1 are tangents drawn from point Р to one ball, and РF 2, РR 2 are tangents drawn from point Р to another ball (Theorem 2 ). Adding both equalities term by term, we find

РF 1 + РF 2 \u003d РR 1 + РR 2 \u003d R 1 R 2 (1)

This relationship shows that the sum of the distances (РF 1 and РF 2) of an arbitrary point P of the ellipse to two points F 1 and F 2 is a constant value for a given ellipse (i.e., it does not depend on the position of point P on the ellipse).

Points F 1 and F 2 are called the focal points of the ellipse. The points at which the line F 1 F 2 intersects the ellipse are called the vertices of the ellipse. The segment between the vertices is called the major axis of the ellipse.

The length of the generatrix R 1 R 2 is equal to the major axis of the ellipse. Then the main property of the ellipse is formulated as follows: the sum of the distances of an arbitrary point P of the ellipse to its foci F 1 and F 2 is a constant value for a given ellipse, equal to the length of its major axis.

Note that if the foci of the ellipse coincide, then the ellipse is a circle, i.e. a circle is a special case of an ellipse.

2.2 Equation of an ellipse

To form the equation of an ellipse, we must consider the ellipse as a locus of points that have some property that characterizes this locus. Let's take the main property of an ellipse as its definition: An ellipse is a locus of points on a plane for which the sum of the distances to two fixed points F 1 and F 2 of this plane, called foci, is a constant value equal to the length of its major axis.

Let the length of the segment F 1 F 2 \u003d 2c, and the length of the major axis is 2a. To derive the canonical equation of the ellipse, we choose the origin O of the Cartesian coordinate system in the middle of the segment F 1 F 2, and direct the axes Ox and Oy as shown in Figure 5. (If the foci coincide, then O coincides with F 1 and F 2, and for axis OX you can take any axis passing through O). Then in the selected coordinate system the points F 1 (s, 0) and F 2 (-s, 0). Obviously, 2a\u003e 2c, i.e. a\u003e c. Let M (x, y) be a point in the plane belonging to an ellipse. Let МF 1 \u003d r 1, МF 2 \u003d r 2. According to the definition of an ellipse, the equality

r 1 + r 2 \u003d 2a (2) is a necessary and sufficient condition for the location of the point M (x, y) on a given ellipse. Using the formula for the distance between two points, we get

r 1 \u003d, r 2 \u003d. Let's return to equality (2):

Let's move one root to the right side of the equality and square it:

Reducing, we get:

We give similar ones, reduce them by 4 and isolate the radical:

Squaring

Expand the brackets and reduce to:

whence we get:

(a 2 -c 2) x 2 + a 2 y 2 \u003d a 2 (a 2 -c 2). (3)

Note that a 2 -c 2\u003e 0. Indeed, r 1 + r 2 is the sum of the two sides of the triangle F 1 MF 2, and F 1 F 2 is its third side. Therefore, r 1 + r 2\u003e F 1 F 2, or 2а\u003e 2с, i.e. a\u003e c. Let us denote a 2 -c 2 \u003d b 2. Equation (3) will have the form: b 2 x 2 + a 2 y 2 \u003d a 2 b 2. Let us perform the transformation that brings the equation of the ellipse to the canonical (literally: taken as a sample) form, namely, divide both sides of the equation by a 2 b 2:

(4) - the canonical equation of the ellipse.

Since equation (4) is an algebraic consequence of equation (2 *), the coordinates x and y of any point M of the ellipse will also satisfy equation (4). Since during algebraic transformations associated with getting rid of radicals, "extra roots" could appear, it is necessary to make sure that any point M whose coordinates satisfy Eq. (4) is located on this ellipse. To do this, it is enough to prove that the values \u200b\u200br 1 and r 2 for each point satisfy relation (2). So, let the coordinates x and y of the point M satisfy equation (4). Substituting the value у 2 from (4) into the expression r 1, after simple transformations we find that r 1 \u003d. Since, then r 1 \u003d. In a completely similar way, we find that r 2 \u003d. Thus, for the considered point М r 1 \u003d, r 2 \u003d, i.e. r 1 + r 2 \u003d 2a, so point M is located on the ellipse. The quantities a and b are called, respectively, the major and minor semiaxes of the ellipse.

2.3 Study of the shape of an ellipse by its equation

Let us establish the shape of the ellipse, using its canonical equation.

1. Equation (4) contains x and y only in even powers, therefore if a point (x, y) belongs to an ellipse, then it also contains points (x, - y), (-x, y), (-x, - y). It follows that the ellipse is symmetric about the Ox and Oy axes, as well as about the point O (0,0), which is called the center of the ellipse.

2. Find the points of intersection of the ellipse with the coordinate axes. Putting y \u003d 0, we find two points A1 (a, 0) and A2 (-a, 0) at which the Ox axis intersects the ellipse. Putting in equation (4) x \u003d 0, we find the points of intersection of the ellipse with the axis Oy: B 1 (0, b) and. B 2 (0, - b) Points A 1, A 2, B 1, B 2 are called the vertices of the ellipse.

3. From equation (4) it follows that each term on the left-hand side does not exceed unity, i.e. inequalities and or and take place. Therefore, all points of the ellipse lie inside the rectangle formed by straight lines,.

4. In equation (4), the sum of non-negative terms and is equal to one. Consequently, as one term increases, the other will decrease, i.e. if x increases, then y decreases and vice versa.

From what has been said it follows that the ellipse has the shape shown in Fig. 6 (oval closed curve).

Note that if a \u003d b, then equation (4) will take the form x 2 + y 2 \u003d a 2. This is the equation of the circle. An ellipse can be obtained from a circle with a radius a, if it is compressed in times along the Oy axis. With this compression, point (x; y) goes to point (x; y 1), where. Substituting circles in the equation, we get the equation of the ellipse:.

Let us introduce one more quantity that characterizes the shape of the ellipse.

The eccentricity of an ellipse is the ratio of the focal length 2c to the length 2a of its major axis.

Eccentricity is usually denoted by e: e \u003d Since c< a, то. Заметив, что c 2 = a 2 - b 2 , находим: , отсюда.

From the last equality, it is easy to obtain a geometric interpretation of the eccentricity of the ellipse. For very small numbers a and b are almost equal, that is, the ellipse is close to a circle. If it is close to one, then the number b is very small compared to the number a, and the ellipse is strongly elongated along the major axis. Thus, the eccentricity of the ellipse characterizes the measure of the elongation of the ellipse.

3. Hyperbola

3.1 The main property of hyperbola

Investigating the hyperbola using constructions similar to those used to study the ellipse, we find that the hyperbola has properties similar to those of the ellipse.

We dissect a straight circular cone by plane b intersecting both of its planes, i.e. parallel to its two generators. In the section, you get a hyperbola. Let us draw plane АSB through the ST axis of the cone, perpendicular to plane b.

Let us inscribe two balls into the cone - one in one cavity, the other in the other, so that each of them touches the conical surface and the secant plane. Let the first ball touch the plane b at point F 1 and touch the conical surface along the circle UґVґ. Let the second ball touch the plane b at point F 2 and touch the conical surface along the circle UV.

Choose an arbitrary point M. on the hyperbola. Draw through it the generator of the cone MS and mark the points d and D at which it touches the first and second balls. Let's connect the point M with the points F 1, F 2, which we will call the foci of the hyperbola. Then МF 1 \u003d Md, since both segments are tangent to the first ball, drawn from the point M. Similarly, МF 2 \u003d MD. Subtracting the second equality term by term, we find

MF 1 -MF 2 \u003d Md-MD \u003d dD,

where dD is a constant value (as a generator of a cone with bases UґVґ and UV), independent of the choice of point M on the hyperbola. Let P and Q denote the points at which the line F 1 F 2 intersects the hyperbola. These points P and Q are called the vertices of the hyperbola. The segment PQ is called the real axis of the hyperbola. In the course of elementary geometry, it is proved that dD \u003d PQ. Therefore, MF 1 -MF 2 \u003d PQ.

If point M will be on that branch of the hyperbola, near which the focus F 1 is located, then MF 2 -MF 1 \u003d PQ. Then we finally get MF 1 -MF 2 \u003d PQ.

The modulus of the difference between the distances of an arbitrary point M of the hyperbola from its foci F 1 and F 2 is a constant value equal to the length of the real axis of the hyperbola.

3.2 Equation of hyperbola

Let's take the main property of a hyperbola as its definition: A hyperbola is a locus of points on a plane for which the modulus of the difference in distances to two fixed points F 1 and F 2 of this plane, called foci, is a constant value equal to the length of its real axis.

Let the length of the segment F 1 F 2 \u003d 2с, and the length of the real axis is equal to 2а. To derive the canonical hyperbola equation, we choose the origin O of the Cartesian coordinate system in the middle of the segment F 1 F 2, and direct the axes Ox and Oy as shown in Figure 5. Then, in the selected coordinate system, the points F 1 (c, 0) and F 2 ( -c, 0). Obviously, 2a<2с, т.е. а<с. Пусть М (х, у) - точка плоскости, принадлежащая гиперболе. Пусть МF 1 =r 1 , МF 2 =r 2 . Согласно определению гиперболы равенство

r 1 -r 2 \u003d 2a (5) is a necessary and sufficient condition for the location of the point M (x, y) on a given hyperbola. Using the formula for the distance between two points, we get

r 1 \u003d, r 2 \u003d. Let's return to equality (5):

Squaring both sides of the equality

(x + c) 2 + y 2 \u003d 4a 2 ± 4a + (x-c) 2 + y 2

Reducing, we get:

2 xs \u003d 4a 2 ± 4a-2 xs

± 4a \u003d 4a 2 -4 xs

a 2 x 2 -2a 2 xc + a 2 c 2 + a 2 y 2 \u003d a 4 -2a 2 xc + x 2 c 2

x 2 (s 2 -a 2) - a 2 y 2 \u003d a 2 (s 2 -a 2) (6)

Note that with 2 -а 2\u003e 0. Let us denote c 2 -а 2 \u003d b 2. Equation (6) will have the form: b 2 x 2 -a 2 y 2 \u003d a 2 b 2. Let us perform a transformation that brings the hyperbola equation to the canonical form, namely, we divide both sides of the equation by a 2 b 2: (7) - the canonical equation of the hyperbola, the quantities a and b are the real and imaginary semiaxes of the hyperbola, respectively.

We must make sure that equation (7), obtained by algebraic transformations of equation (5 *), has not acquired new roots. For this, it is sufficient to prove that for each point M, the coordinates x and y of which satisfy equation (7), the values \u200b\u200br 1 and r 2 satisfy relation (5). Carrying out arguments similar to those that were made when deriving the ellipse formula, we find the following expressions for r 1 and r 2:

Thus, for the considered point M we have r 1 -r 2 \u003d 2a, and therefore it is located on the hyperbola.

3.3 Study of the hyperbola equation

Now let us try, on the basis of considering equation (7), to form an idea of \u200b\u200bthe location of the hyperbola.
1. First of all, equation (7) shows that the hyperbola is symmetric with respect to both axes. This is due to the fact that only even powers of coordinates are included in the curve equation. 2. Let us now mark the area of \u200b\u200bthe plane where the curve will lie. The hyperbola equation resolved with respect to y has the form:

It shows that y always exists when x 2? a 2. This means that for x? a and for x? - and the ordinate y will be real, and for - a

Further, with x increasing (and greater than a), the ordinate of y will also grow all the time (in particular, this shows that the curve cannot be wavy, i.e. such that with increasing abscissa x the ordinate of y either increases or decreases) ...

H. The center of a hyperbola is a point relative to which each point of the hyperbola has a point symmetric to itself on it. The point O (0,0), the origin of coordinates, as for the ellipse, is the center of the hyperbola given by the canonical equation. This means that each point of the hyperbola has a symmetric point on the hyperbola with respect to the point O. This follows from the symmetry of the hyperbola with respect to the Ox and Oy axes. Any chord of a hyperbola passing through its center is called the diameter of the hyperbola.

4. The points of intersection of the hyperbola with the straight line on which its foci lie are called the vertices of the hyperbola, and the segment between them is called the real axis of the hyperbola. In this case, the real axis is the Ox axis. Note that the real axis of the hyperbola is often called both the segment 2a and the line itself (the Ox axis) on which it lies.

Let's find the points of intersection of the hyperbola with the Oy axis. The Oy axis equation has the form x \u003d 0. Substituting x \u003d 0 into equation (7), we obtain that the hyperbola has no points of intersection with the Oy axis. This is understandable, since there are no hyperbola points in the 2a-wide strip covering the Oy axis.

A straight line perpendicular to the real axis of the hyperbola and passing through its center is called the imaginary axis of the hyperbola. In this case, it coincides with the Oy axis. So, the denominators of the terms with x 2 and y 2 in the hyperbola equation (7) are the squares of the real and imaginary semiaxes of the hyperbola.

5. The hyperbola intersects with the line y \u003d kx for k< в двух точках. Если k то общих точек у прямой и гиперболы нет.

Evidence

To determine the coordinates of the intersection points of the hyperbola and the straight line y \u003d kx, you need to solve the system of equations

Eliminating y, we get

or When b 2 -k 2 a 2 0 that is, when k is the resulting equation, and therefore the system of solutions does not have.

Lines with equations y \u003d and y \u003d - are called hyperbola asymptotes.

For b 2 -k 2 a 2\u003e 0 that is, for k< система имеет два решения:

Consequently, each straight line passing through the origin with the slope k< пересекает гиперболу в двух точках. При k = 0 получаем точки пересечения (a; 0) и (- a; 0) - вершины гиперболы.

6. Optical property of hyperbola: optical rays emanating from one focus of the hyperbola, reflected from it, seem to emanate from the second focus.

The eccentricity of a hyperbola is the ratio of the focal length 2c to the length 2a of its real axis? \u003d Since c\u003e a, then e\u003e 1, then the foci of the hyperbola, as in the case of an ellipse, is inside the curve,
those. from the side of its concavity.

3.4 Conjugate hyperbola

Along with hyperbola (7), the so-called hyperbola conjugate with respect to it is considered. The conjugate hyperbola is defined by the canonical equation.

In fig. 10 shows the hyperbola (7) and the conjugate hyperbola. The conjugate hyperbola has the same asymptotes as this one, but F 1 (0, c),

4. Parabola

4.1 The main property of a parabola

Let us establish the main properties of the parabola. We cut a straight circular cone with apex S by a plane parallel to one of its generators. In the section we get a parabola. Let us draw plane АSB through the ST axis of the cone, perpendicular to the plane (Fig. 11). The generatrix SA lying in it will be parallel to the plane. Let us inscribe into the cone a spherical surface tangent to the cone along the circle UV and tangent to the plane at point F. Draw through the point F a straight line parallel to the generator SA. Let us denote the point of its intersection with the generator SB by P. Point F is called the focus of the parabola, point P is its apex, and the line PF passing through the vertex and focus (and parallel to the generator SA) is called the parabola axis. The parabola will not have the second vertex - the point of intersection of the РF axis with the generatrix SA: this point “goes to infinity”. Let us call the directrix (translated as “guide”) the line q 1 q 2 of the intersection of the plane with the plane in which the circle UV lies. Take an arbitrary point M on the parabola and connect it to the vertex of the cone S. The line MS touches the ball at point D lying on the circle UV. Connect point M with focus F and drop the perpendicular MK to the directrix from point M. Then it turns out that the distances of an arbitrary point M of the parabola to the focus (MF) and to the directrix (MK) are equal to each other (the main property of the parabola), i.e. MF \u003d MK.

Proof: МF \u003d MD (as tangents to the ball from one point). Let us denote the angle between any of the generatrices of the cone and the ST axis through c. We will project the segments MD and MK onto the ST axis. The segment MD forms a projection on the ST axis, equal to MDcosc, since MD lies on the generatrix of the cone; the MK segment forms a projection on the ST axis, equal to MKsots, since the MK segment is parallel to the generatrix SA. (Indeed, the directrix q 1 q 1 is perpendicular to the plane ASB. Consequently, the straight line РF intersects the directrix at the point L. At the right angle. But the lines MK and PF lie in the same plane, and MK is also perpendicular to the directrix). The projections of both segments MK and MD on the ST axis are equal to each other, since one of their ends - point M - is common, and the other two D and K lie in a plane perpendicular to the ST axis (Fig.). Then MDcosts \u003d MKsots or MD \u003d MK. Therefore, MF \u003d MK.

Property 1. (Focal property of a parabola).

The distance from any point of the parabola to the middle of the main chord is equal to its distance to the directrix.

Evidence.

Point F is the intersection point of the line QR and the main chord. This point lies on the axis of symmetry Oy. Indeed, triangles RNQ and ROF are equal, as rectangular

triangles with wound legs (NQ \u003d OF, OR \u003d RN). Therefore, no matter what point N we take, the straight line QR constructed from it will intersect the main chord in its middle F. Now it is clear that the triangle FMQ is isosceles. Indeed, the segment MR is both the median and the height of this triangle. Hence it follows that MF \u003d MQ.

Property 2. (Optical property of a parabola).

Any tangent to a parabola makes equal angles with a focal radius drawn to the point of tangency, and a ray passing from the point of tangency and co-directional with the axis (or, rays coming out of a single focus, reflected from the parabola, will go parallel to the axis).

Evidence. For a point N lying on the parabola itself, the equality | FN | \u003d | NH | is true, and for a point N "lying in the inner region of the parabola, | FN" |<|N"H"|. Если теперь провести биссектрису l угла FМК, то для любой отличной от М точки M" прямой l найдём:

| FM "| \u003d | M" K "|\u003e | M" K "|, that is, point M" lies in the outer region of the parabola. So, the whole line l, except for the point M, lies in the outer region, that is, the inner region of the parabola lies on one side of l, which means that l is tangent to the parabola. This gives proof of the optical property of the parabola: angle 1 is equal to angle 2, since l is the bisector of the angle FMK.

4.2 Equation of a parabola

Based on the main property of a parabola, let us formulate its definition: a parabola is the set of all points of the plane, each of which is equally distant from a given point, called the focus, and a given straight line, called the directrix. The distance from the focus F to the directrix is \u200b\u200bcalled the parameter of the parabola and is denoted by p (p\u003e 0).

To derive the parabola equation, we choose the Oxy coordinate system so that the Ox axis passes through the focus F perpendicular to the directrix in the direction from the directrix to F, and the origin of coordinates O is located in the middle between the focus and the directrix (Fig. 12). In the chosen system, the focus is F (, 0), and the directrix equation has the form x \u003d -, or x + \u003d 0 Let m (x, y) be an arbitrary point of the parabola. Let's connect the point M with F. Let's draw the segment MH perpendicular to the directrix. According to the definition of the parabola MF \u003d MH. Using the formula for the distance between two points, we find:

Therefore, Squaring both sides of the equation, we get

those. (8) Equation (8) is called the canonical equation of the parabola.

4.3 Study of the shape of a parabola by its equation

1. In equation (8) the variable y is included in an even power, which means that the parabola is symmetric about the Ox axis; the Ox axis is the axis of symmetry of the parabola.

2. Since c\u003e 0, it follows from (8) that x\u003e 0. Consequently, the parabola is located to the right of the Oy axis.

3. Let x \u003d 0, then y \u003d 0. Therefore, the parabola passes through the origin.

4. With an unbounded increase in x, the module у also increases unboundedly. The parabola y 2 \u003d 2 px has the form (shape) shown in Figure 13. Point O (0; 0) is called the apex of the parabola, the segment FM \u003d r is called the focal radius of point M. Equations y 2 \u003d -2 px, x 2 \u003d - 2 py, x 2 \u003d 2 py (p\u003e 0) also define parabolas.

1.5. Directory property of conic sections .

Here we will prove that each non-circular (non-degenerate) conic section can be defined as a set of points M, the ratio of the distance MF of which from a fixed point F to the distance MP from a fixed straight line d not passing through the point F is equal to a constant value e: where F is the focus of the conical section, line d is the directrix, and the ratio e is the eccentricity. (If a point F belongs to the straight line d, then the condition defines a set of points, which is a pair of straight lines, i.e., a degenerate conical section; for e \u003d 1, this pair of straight lines merges into one straight line. For the proof, consider the cone formed by the rotation of the straight line l around the intersecting its at point O of the straight line p, making angle b< 90є; пусть плоскость р не проходит через вершину конуса и образует с его осью p угол в < 90є (если в = 90є, то плоскость р пересекает конус по окружности).

We inscribe into the cone a ball K tangent to the plane p at point F and tangent to the cone along the circle S. The line of intersection of the plane p with the plane at the circle S is denoted by d.

Now we connect an arbitrary point M lying on the line L of the intersection of the plane p and the cone with the vertex O of the cone and with the point F and drop from M the perpendicular MP to the line d; we also denote by E the point of intersection of the generator MO of the cone with the circle S.

Moreover, MF \u003d ME, as the segments of two tangent lines of the ball K drawn from one point M.

Further, the segment ME forms a constant angle b with the axis p of the cone (that is, independent of the choice of point M), and the segment MP forms a constant angle b; therefore, the projections of these two segments on the p axis are respectively equal to ME cos b and MP cos c.

But these projections coincide, since the segments ME and MP have a common origin M, and their ends lie in the y plane, perpendicular to the p axis.

Therefore, ME cos b \u003d MP cos c, or, since ME \u003d MF, MF cos b \u003d MP cos c, whence it follows that

It is also easy to show that if the point M of the plane p does not belong to the cone, then. Thus, each section of a right circular cone can be described as a set of points in the plane for which. On the other hand, by changing the values \u200b\u200bof the angles b and c, we can give the eccentricity any value e\u003e 0; further, from considerations of similarity, it is easy to understand that the distance FQ from the focus to the directrix is \u200b\u200bdirectly proportional to the radius r of the ball K (or the distance d of the plane p from the vertex O of the cone). It can be shown that, thus, by choosing a suitable distance d, we can give the distance FQ any value. Therefore, each set of points M, for which the ratio of the distances from M to a fixed point F and to a fixed straight line d has a constant value, can be described as a curve obtained in the section of a straight circular cone by a plane. This proves that (non-degenerate) conic sections can also be defined by the property referred to in this subsection.

This property of conic sections is called them directory property... It is clear that if c\u003e b, then e< 1; если в = б, то е = 1; наконец, если в < б, то е > 1. On the other hand, it is easy to see that if c\u003e b, then the plane p intersects the cone along a closed bounded line; if c \u003d b, then the plane p intersects the cone along an unbounded line; if in< б, то плоскость р пересекает обе полы конуса и, следовательно, линия пересечения этой плоскости и конуса состоит из двух (неограниченных) частей или ветвей (рис. 17).

Conical section for which e< 1, называется эллипсом; коническое сечение с эксцентриситетом е = 1 называется параболой; коническое сечение, для которого е > 1 is called a hyperbola. Ellipses also include a circle, which cannot be specified by a directory property; since for a circle the ratio turns to 0 (since in this case b \u003d 90є), it is conventionally considered that the circle is a conical section with an eccentricity of 0.

6. Ellipse, hyperbola and parabola as conical sections

conical section ellipse hyperbola

The ancient Greek mathematician Menechm, who discovered the ellipse, hyperbola and parabola, defined them as sections of a circular cone by a plane perpendicular to one of the generators. He called the resulting curves sections of acute-angled, rectangular and obtuse-angled cones, depending on the axial angle of the cone. The first, as we will see below, is an ellipse, the second is a parabola, and the third is one branch of a hyperbola. The names "ellipse", "hyperbola" and "parabola" were introduced by Apollonius. Almost completely (7 out of 8 books) of Apollonius's composition "On conical sections" has come down to us. In this work, Apollonius examines both sides of the cone and intersects the cone with planes not necessarily perpendicular to one of the generatrices.

Theorem. The section of any straight round cone by a plane (not passing through its vertex) defines a curve that can only be a hyperbola (Fig. 4), a parabola (Fig. 5) or an ellipse (Fig. 6). Moreover, if the plane intersects only one plane of the cone and along a closed curve, then this curve is an ellipse; if the plane intersects only one plane along an open curve, then this curve is a parabola; if the cutting plane intersects both planes of the cone, then a hyperbola is formed in the section.

An elegant proof of this theorem was proposed in 1822 by Dandelen using spheres, which are now commonly called Dandelen spheres. Consider this proof.

Let us inscribe into the cone two spheres tangent to the plane of the section П from different sides. Let F1 and F2 denote the tangency points of this plane with the spheres. Let us take an arbitrary point M. on the line of section of the cone by plane P. Note on the generatrix of the cone passing through M, points P1 and P2, lying on the circle k1 and k2, along which the spheres touch the cone.

It is clear that МF1 \u003d МР1 as segments of two tangents to the first sphere going out from М; similarly, МF2 \u003d МР2. Therefore, MF1 + MF2 \u003d MP1 + MP2 \u003d P1P2. The length of the segment P1P2 is the same for all points M of our section: this is the generatrix of the truncated cone bounded by parallel planes 1 and 11, in which the circles k1 and k2 lie. Consequently, the line of section of the cone by plane P is an ellipse with foci F1 and F2. The validity of this theorem can also be established proceeding from the general position that the intersection of a second-order surface by a plane is a second-order line.

Literature

1. Atanasyan L.S., Bazylev V.T. Geometry. In 2 hours. Part 1. Textbook for students of physics and mathematics. ped. in - comrade-M .: Education, 1986.

2. Bazylev V.T. and other Geometry. Textbook. manual for 1st year students nat. - mat. facts - tov ped. in. - Comrade-M .: Education, 1974.

3. Pogorelov A.V. Geometry. Textbook. for 7-11 cl. Wednesday shk. - 4th ed.-M .: Education, 1993.

4. The history of mathematics from ancient times to the beginning of the 19th century. A.P. Yushkevich - Moscow: Nauka, 1970.

5. Boltyansky V.G. Optical properties of ellipse, hyperbola and parabola. // Quant. - 1975. - No. 12. - with. 19 - 23.

6. Efremov N.V. A short course in analytical geometry. - M: Science, 6th edition, 1967. - 267 p.

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The main sections of the cone. Section formed by a plane passing through the axis of the cone (axial) and through its apex (triangle). Formation of a section by a plane parallel (parabola), perpendicular (circle) and not perpendicular (ellipse) axis.

Municipal educational institution

Alekseevskaya secondary school

"Education Centre"

Lesson development

Topic: STRAIGHT CIRCULAR CONE.

SECTION OF THE CONE BY PLANES

Mathematic teacher

academic year

Topic: STRAIGHT CIRCULAR CONE.

SECTION OF THE CONE BY PLANES.

The purpose of the lesson:parse the definitions of the cone and subordinate concepts (top, base, generators, height, axis);

consider the sections of the cone passing through the apex, including axial sections;

contribute to the development of the spatial imagination of students.

Lesson objectives:

Educational: study the basic concepts of a body of revolution (cone).

Developing: continue to develop the skills of analysis, comparison; skills to highlight the main thing, to formulate conclusions.

Educational: fostering students' interest in learning, instilling communication skills.

Lesson type:lecture.

Teaching methods:reproductive, problematic, partly exploratory.

Equipment:table, models of bodies of rotation, multimedia equipment.

During the classes

I. Organizing time.

In the previous lessons, we already got acquainted with bodies of revolution and dwelled on the concept of a cylinder in more detail. On the table you see two drawings and working in pairs, formulate the correct questions on the topic covered.

P. Checking homework.

Work in pairs using a thematic table (a prism inscribed in a cylinder and a prism described near a cylinder).

For example, in pairs and individually, students may ask questions:

What is a circular cylinder (generatrix of a cylinder, base of a cylinder, side surface of a cylinder)?

What prism is called described near the cylinder?

Which plane is called the tangent to the cylinder?

What shapes can be called polygons ABC, A1 B1 C1 , ABCDE andA1 B1 C1 D1 E1 ?

- What prism is a prism ABCDEABCDE? (Straightmy.)

- Prove that it is a straight prism.

(optional, 2 pairs of students at the blackboard do the work)

III. Updating basic knowledge.

By planimetry material:

Thales' theorem;

Triangle centerline properties;

Area of \u200b\u200ba circle.

By stereometry material:

Concept homothety;

The angle between a straight line and a plane.

IV.Learning new material.

(educational - methodical set "Living Mathematics », attachment 1.)

After the presented material, a work plan is proposed:

1. Definition of the cone.

2. Definition of a straight cone.

3. Elements of the cone.

4. Development of the cone.

5. Obtaining a cone as a body of revolution.

6. Types of sections of the cone.

Students independently find the answers to these questionschildren in paragraphs 184-185, accompanying them with drawings.

Valeological pause:Are you tired? Let's take some rest before the next practical stage of work!

Massage of reflex zones on the auricle responsible for work internal organs;

· Massage of reflex zones on the palms of the hands;

· Gymnastics for the eyes (close your eyes and sharply open your eyes);

Spine stretch (raise your arms up, pull yourself up with your right and then your left arm)

Respiratory gymnastics aimed at saturating the brain with oxygen (inhale sharply through the nose 5 times)

A thematic table is compiled (together with the teacher), accompanying the filling of the table with questions and material received from various sources (textbook and computer presentation)

"Cone. Frustum".

Thematic table 1. Cone (straight, circular) is called a body obtained by rotating a right-angled triangle around a straight line containing a leg. Dot M - vertex cone, circle with center ABOUT – basecone, line segment MA=l aboutdestructive cone, segment MO= H - cone height, line segment OA= R - base radius, segment Sun= 2 R - base diametervania, triangle MVS -axial section, < BMC - angle at the top of the axial section, < MBO - angleslope of the generatrix to the planebase bones _________________________________________ 2. Cone sweep - sector circle and circle. < BMBl = and - sweep angle... Sweep arc length ВСВ1 \u003d 2π R = la . Lateral surface area S lateral. \u003d π R l Total surface area (sweep area) S \u003d π R ( l + R )	Cone called a body that consists of a circle - grounds a cone, a point not lying in the plane of this circle - tops cone and all line segments connecting the top of the cone with the base points - generators ______________________________
3. Sections of the cone by planes
Section of a cone by a plane passing through the top of the cone, - isosceles triangle AMB: AM \u003d BM - generators of the cone, AB - chord; Axial section- isosceles triangle AMB: AM \u003d BM - generators of the cone, AB - base diameter.
Section of the cone by a plane perpendicular to the axis of the cone, - a circle; at an angle to the axis of the cone - ellipse.
Truncated cone is called the part of the cone enclosed between the base and the section of the cone parallel to the base. Circles with centers 01 and O2 - top and bottom bases truncated cone, r andR - base radii, line segment AB= l - generator, ά - tilt angleto the plane lower base, line segment 01O2 -height(distance between flatgrounds), trapezoid ABCD - axial section.

V.Securing the material.

Frontal work.

· Verbally (using the finished drawing) No. 9 and No. 10 are being solved.

(two students explain the solution to problems, the rest can take short notes in notebooks)

No. 9. The radius of the base of the cone is 3m, the height of the cone is 4m. find the generator.

(Decision:l=√ R2 + H2 \u003d √32 + 42 \u003d √25 \u003d 5m.)

No. 10 Generator of the cone l inclined to the plane of the base at an angle of 30 °. Find the height.

(Decision:H = l sin30◦ = l|2.)

· Solve the problem on the finished drawing.

The height of the cone is h. Through generators MAand MB a plane is drawn making an angle andwith the plane of the base of the cone. Chord ABconstricts the arc with the degree measure r.

1. Prove that the section of the cone by the plane MAV- isosceles triangle.

2. Explain how to construct the linear angle of the dihedral formed by the cutting plane and the plane of the base of the cone.

3. Find MS.

4. Make (and explain) a plan for calculating the chord length ABand cross-sectional area MAV.

5. Show in the figure how you can draw a perpendicular from a point ABOUTto the section plane MAV(justify the construction).

· Reiteration:

studied material from planimetry:

Definition of an isosceles triangle;

Properties of an isosceles triangle;

Area of \u200b\u200ba triangle

of the studied material from stereometry:

Determination of the angle between the planes;

A method for constructing a linear angle of a dihedral angle.

Self-test test

1. Draw bodies of revolution formed by rotating the plane shapes shown in the figure.

2. Specify which rotation flat figure the depicted body of rotation turned out. (b)

The diagnostic work consists of two parts, including 19 tasks. Part 1 contains 8 tasks basic level difficulty with a short answer. Part 2 contains 4 tasks of an increased level of difficulty with a short answer and 7 tasks of an increased and high level difficulties with a detailed answer.
For execution diagnostic work in mathematics, 3 hours 55 minutes (235 minutes) are allotted.
Answers to tasks 1-12 are written as an integer or final decimal fraction. Write the numbers in the answer fields in the text of the work, and then transfer them to answer form No. 1. When completing tasks 13-19, you need to write down the complete solution and answer in answer form No. 2.
All forms are filled in with bright black ink. The use of gel, capillary or fountain pens is allowed.
When completing assignments, you can use the draft. Draft entries do not count towards grading work.
The points received by you for completed tasks are summed up.
We wish you success!

Problem conditions

1. Find if
2. To obtain an enlarged image of a light bulb on the screen, a collecting lens with a main focal length \u003d 30 cm is used in the laboratory. The distance from the lens to the bulb can vary from 40 to 65 cm, and the distance from lens to screen - within the range from 75 to 100 cm. The image on the screen will be clear if the ratio is met. Indicate on which greatest distance a light bulb can be placed from the lens so that its image on the screen is clear. Express your answer in centimeters.
3. The motor ship goes along the river to its destination 300 km and after stopping returns to the point of departure. Find the speed of the current, if the speed of the ship in still water is 15 km / h, the stay lasts 5 hours, and the ship returns to the point of departure 50 hours after sailing from it. Give your answer in km / h.
4. Find the smallest function value on the segment
5. a) Solve the equation b) Find all the roots of this equation belonging to the segment
6. Given a straight circular cone with apex M... The axial section of the cone is a triangle with an angle of 120 ° at the apex M... The generatrix of the cone is. Through point M the section of the cone is drawn, perpendicular to one of the generators.
   a) Prove that the resulting triangle in the section is obtuse.
   b) Find the distance from the center ABOUT the base of the cone to the section plane.
7. Solve the equation
8. Circle with center ABOUTtouches the side ABisosceles triangle ABC,side extensions ASand continuing the foundation Sunat the point N... Dot M- middle of the base Sun.
   a) Prove that MN \u003d AC.
   b) Find OS,if the sides of the triangle ABCare 5, 5 and 8.
9. Business project "A" assumes an increase in the amount invested in it by 34.56% annually during the first two years and by 44% annually over the next two years. Project "B" assumes growth by a constant integer n percent annually. Find the smallest value n, in which in the first four years project "B" will be more profitable than project "A".
10. Find all values \u200b\u200bof the parameter,, for each of which the system of equations has the only solution
11. Anya plays a game: two different natural numbers are written on the board and, both are less than 1000. If both are natural, then Anya makes a move - replaces the previous ones with these two numbers. If at least one of these numbers is not natural, then the game is over.
    a) Can the game continue for exactly three moves?
    b) Are there two initial numbers such that the game will last at least 9 moves?
    c) Anya made the first move in the game. Find the largest possible ratio of the product of the two numbers obtained to the product

Let a straight circular cylinder be given, the horizontal plane of projections is parallel to its base. When the plane intersects the cylinder general position (we assume that the plane does not intersect the bases of the cylinder) the intersection line is an ellipse, the section itself has the shape of an ellipse, its horizontal projection coincides with the projection of the base of the cylinder, and the frontal one also has the shape of an ellipse. But if the section plane makes an angle of 45 ° with the cylinder axis, then the elliptical section is projected by a circle onto the projection plane to which the section is inclined at the same angle.

If the cutting plane intersects the lateral surface of the cylinder and one of its bases (Fig. 8.6), then the intersection line has the shape of an incomplete ellipse (part of an ellipse). The horizontal projection of the section in this case is part of the circle (projection of the base), and the frontal projection is part of the ellipse. The plane can be located perpendicular to any projection plane, then the section will be projected onto this projection plane by a straight line (part of the trail of the secant plane).

If the cylinder is intersected by a plane parallel to the generatrix, then the lines of intersection with the lateral surface are straight, and the section itself has the shape of a rectangle if the cylinder is straight, or a parallelogram if the cylinder is inclined.

As is known, both the cylinder and the cone are formed by ruled surfaces.

The line of intersection (cut line) of the ruled surface and the plane in the general case is a certain curve, which is constructed from the points of intersection of the generatrices with the cutting plane.

Let it be given straight circular cone. When it intersects with a plane, the intersection line can have the shape of a triangle, ellipse, circle, parabola, hyperbola (Fig. 8.7), depending on the location of the plane.

A triangle is obtained when the cutting plane, crossing the cone, passes through its vertex. In this case, the lines of intersection with the side surface are straight lines intersecting at the apex of the cone, which, together with the line of intersection of the base, form a triangle projected on the projection plane with distortion. If the plane intersects the axis of the cone, then a triangle is obtained in the section, in which the angle with the apex coinciding with the apex of the cone will be the maximum for the sections-triangles this cone... In this case, the section is projected onto the horizontal projection plane (it is parallel to its base) by a straight line segment.

The line of intersection of the plane and the cone will be an ellipse if the plane is not parallel to any of the generatrices of the cone. This is equivalent to the fact that the plane intersects all generators (the entire lateral surface of the cone). If the cutting plane is parallel to the base of the cone, then the intersection line is a circle, the section itself is projected onto the horizontal projection plane without distortion, and onto the frontal plane - by a straight line segment.

The line of intersection will be parabolic when the cutting plane is parallel to only one generatrix of the cone. If the secant plane is parallel to two generators simultaneously, then the intersection line is a hyperbola.

A truncated cone is obtained if a straight circular cone is intersected by a plane parallel to the base and perpendicular to the axis of the cone, and the upper part is discarded. In the case when the horizontal projection plane is parallel to the bases of the truncated cone, these bases are projected onto the horizontal projection plane without distortion by concentric circles, and the frontal projection is a trapezoid. When a plane intersects a truncated cone, depending on its location, the cut line can have the shape of a trapezoid, ellipse, circle, parabola, hyperbola, or part of one of these curves, the ends of which are connected by a straight line.