Municipal Educational Institution

Average Comprehensive school №4

Conic sections

Completed

Spiridonov Anton

student of class 11A

Checked

Korobeynikova A. T.

Tobolsk - 2006

Introduction

The concept of conic sections

Types of conic sections

Study

Construction of conic sections

Analytical approach

Application

References

Introduction.

Purpose: to study conic sections.

Objectives: learn to distinguish between types of conic sections, construct kinetic sections and apply an analytical approach.

Conic sections were first proposed to be used by the ancient Greek geometer Menaechmus, who lived in the 4th century BC, when solving the problem of doubling a cube. This task is associated with the following legend.

One day, a plague epidemic broke out on the island of Delos. The inhabitants of the island turned to the oracle, who said that to stop the epidemic it was necessary to double the golden altar, which had the shape of a cube and was located in the temple of Apollo in Athens. The islanders made a new altar, the ribs of which were twice as large as the ribs of the previous one. However, the plague did not stop. The angry residents heard from the oracle that they misunderstood his instructions - it was not the edges of the cube that needed to be doubled, but its volume, that is, the edges of the cube should be increased by

once. In terms geometric algebra, which was used by Greek mathematicians, the problem meant: according to this segment and find segments x and y such that a: x = x: y = y: 2a. Then the length of the segment x will be equal to .

The given proportion can be considered as a system of equations:

But x 2 =ay and y 2 =2ax are equations of parabolas. Therefore, to solve the problem, one must find their intersection points. If we take into account that the equation of the hyperbola xy=2a 2 can also be obtained from the system, then the same problem can be solved by finding the points of intersection of the parabola and the hyperbola.

To obtain conic sections, Menaechmus intersected a cone - acute, rectangular or obtuse - with a plane perpendicular to one of the generatrices. For an acute-angled cone, the section by a plane perpendicular to its generatrix has the shape of an ellipse. An obtuse cone gives a hyperbola, and a rectangular cone gives a parabola.

This is where the names of curves come from, which were introduced by Apollonius of Perga, who lived in the 3rd century BC: ellipse (έλλείψίς), which means a flaw, a deficiency (of the angle of a cone to a straight line); hyperbola (ύπέρβωλη) - exaggeration, preponderance (of a cone angle over a straight line); parabola (παραβολη) - approximation, equality (of the cone angle right angle). Later the Greeks noticed that all three curves could be obtained on one cone by changing the inclination of the cutting plane. In this case, you should take a cone consisting of two cavities and think that they extend to infinity (Fig. 1).

If you make a section circular cone, perpendicular to its axis, and then rotate the cutting plane, leaving one point of its intersection with the cone motionless, we will see how the circle will first stretch out, turning into an ellipse. Then the second vertex of the ellipse will go to infinity, and instead of an ellipse you will get a parabola, and then the plane will also intersect the second cavity of the cone and you will get a hyperbola.

The concept of conic sections.

Conic sections are plane curves that are obtained by intersecting a right circular cone with a plane that does not pass through its vertex. From the point of view analytical geometry conic section is locus points satisfying the second order equation. With the exception of degenerate cases discussed in the last section, conic sections are ellipses, hyperbolas or parabolas (Fig. 2).

When rotating right triangle near one of the legs, the hypotenuse with its extensions describes a conical surface, called the surface of a right circular cone, which can be considered as a continuous series of lines passing through the vertex and called generators, all generators resting on the same circle, called the generator. Each of the generators represents the hypotenuse of a rotating triangle (in its known position), extended in both directions to infinity. Thus, each generatrix extends on both sides of the vertex, as a result of which the surface has two cavities: they converge at one point at common top. If such a surface is intersected by a plane, then the section will produce a curve, which is called a conic section. It can be of three types:

1) if a plane intersects a conical surface along all generatrices, then only one cavity is dissected and a closed curve called an ellipse is obtained in the section;

2) if the cutting plane intersects both cavities, then a curve is obtained that has two branches and is called a hyperbola;

3) if the cutting plane is parallel to one of the generatrices, then a parabola is obtained.

If the cutting plane is parallel to the generating circle, then a circle is obtained, which can be considered as special case ellipse. A cutting plane can intersect a conical surface only at one vertex, then the section produces a point, as a special case of an ellipse.

If a plane passing through the vertex intersects both cavities, then the section produces a pair of intersecting lines, considered as a special case of a hyperbola.

If a vertex is at infinity, then conical surface turns into a cylindrical, and its section is a plane, parallel to the generators, gives a pair of parallel lines as a special case of a parabola. Conic sections are expressed by 2nd order equations, the general form of which is

Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0

and are called 2nd order curves.

Types of conic sections.

Conic sections can be three types:

1) the cutting plane intersects all generatrices of the cone at points of one of its cavity; the intersection line is a closed oval curve - an ellipse; a circle as a special case of an ellipse is obtained when the cutting plane is perpendicular to the axis of the cone.

2) The cutting plane is parallel to one of the tangent planes of the cone; in cross-section, the result is an open curve that goes to infinity - a parabola, lying entirely on one cavity.

3) The cutting plane intersects both cavities of the cone; the line of intersection - a hyperbola - consists of two identical open parts extending to infinity (branches of the hyperbola) lying on both cavities of the cone.

Study.

In cases where a conic section has a center of symmetry (center), i.e., is an ellipse or hyperbola, its equation can be reduced (by moving the origin of coordinates to the center) to the form:

a 11 x 2 +2a 12 xy + a 22 y 2 = a 33 .

Further studies of such (called central) conic sections show that their equations can be reduced to an even simpler form:

Ax 2 + Wu 2 = C,

if we choose the main directions for the directions of the coordinate axes - the directions of the main axes (axes of symmetry) of conic sections. If A and B have identical signs(coinciding with the sign C), then the equation defines an ellipse; if A and B different sign, then - a hyperbole.

The equation of a parabola cannot be reduced to the form (Ax 2 + By 2 = C). With a proper choice of coordinate axes (one coordinate axis is the only axis of symmetry of the parabola, the other is a straight line perpendicular to it, passing through the vertex of the parabola), its equation can be reduced to the form:

CONSTRUCTION OF CONIC SECTIONS.

Studying conic sections as intersections of planes and cones, ancient Greek mathematicians also considered them as trajectories of points on a plane. It was found that an ellipse can be defined as the locus of points, the sum of the distances from which to two given points is constant; parabola - as a locus of points equidistant from given point and a given straight line; hyperbola - as a locus of points, the difference in distances from which to two given points is constant.

These definitions of conic sections as plane curves also suggest a method for constructing them using a stretched string.

Ellipse. If the ends of a thread of a given length are fixed at points F 1 and F 2 (Fig. 3), then the curve described by the point of a pencil sliding along a tightly stretched thread has the shape of an ellipse. Points F 1 and F 2 are called the foci of the ellipse, and the segments V 1 V 2 and v 1 v 2 between the points of intersection of the ellipse with the coordinate axes - the major and minor axes. If points F 1 and F 2 coincide, then the ellipse turns into a circle (Fig. 3).

Hyperbola. When constructing a hyperbola, point P, the tip of a pencil, is fixed on a thread that slides freely along pegs installed at points F 1 and F 2, as shown in Figure 4, a, the distances are selected so that segment PF 2 is longer than segment PF 1 by a fixed value less than the distance F 1 F 2 . In this case, one end of the thread passes under the peg F 1, and both ends of the thread pass over the peg F 2. (The point of the pencil should not slide along the thread, so it must be secured by making a small loop on the thread and threading the point through it.) We draw one branch of the hyperbola (PV 1 Q), making sure that the thread remains taut all the time, and, pulling both ends of the thread down past point F 2, and when point P is below the segment F 1 F 2, holding the thread by both ends and carefully releasing it. We draw the second branch of the hyperbola by first changing the pins F 1 and F 2 (Fig. 4).

When a conical surface is sectioned by a plane, curves of the second order are obtained - circle, ellipse, parabola and hyperbola. In a frequent case, at a certain location of the cutting plane and when it passes through the vertex of the cone (S∈γ), the circle and ellipse degenerate into a point or one or two generatrices of the cone fall into the section.

Gives - a circle when the cutting plane is perpendicular to its axis and intersects all generating surfaces.

Gives - an ellipse when the cutting plane is not perpendicular to its axis and intersects all generating surfaces.

Let's construct an elliptic ω plane α , occupying a general position.

Solving the problem on cross section of a right circular cone plane is greatly simplified if the cutting plane occupies a projecting position.

Using the method of changing projection planes, we translate the plane α from general position in particular - front-projecting. On frontal plane projections V 1 let's build a trace of the plane α and projection of the cone surface ω plane gives an ellipse, since the cutting plane intersects all the generatrices of the cone. The ellipse is projected onto the projection plane as a second-order curve.
On the trace of the plane α V take an arbitrary point 3" we measure its distance from the projection plane H and lay it down along the communication line already on the plane V 1, getting a point 3" 1 . The trail will pass through it αV 1. Cone section line ω - points A" 1, E" 1 coincides here with the trace of the plane. Next, we will construct an auxiliary cutting plane γ3, drawing on the frontal plane of projections V 1 her trail γ 3V 1. Auxiliary plane intersecting a conical surface ω will give a circle, and intersecting with a plane α will give a horizontal line h3. In turn, the straight line intersecting with the circle gives the required points C` and K` plane intersections α with conical surface ω . Frontal projections of the required points C" and K" construct as points belonging to the cutting plane α .

To find a point E(E`, E") section lines, draw a horizontally projecting plane through the top of the cone γ 2 H, which will intersect the plane α in a straight line 1-2(1`-2`, 1"-2") . Intersection 1"-2" with the communication line gives a point E" - highest point section lines.

To find the point indicating the visibility limit of the frontal projection of the section line, draw a horizontally projecting plane through the top of the cone γ 5 H and find the horizontal projection F` the desired point. Also, plane γ 5 H will intersect the plane α frontal f(f`, f"). Intersection f" with the communication line gives a point F". We connect the points obtained on the horizontal projection with a smooth curve, marking on it the leftmost point G - one of characteristic points intersection lines.
Then, we construct projections G on the frontal planes of projections V1 and V. We connect all the constructed points of the section line on the frontal plane of projections V with a smooth line.

Gives - a parabola when the cutting plane is parallel to one generatrix of the cone.

When constructing projections of curves - conic sections, it is necessary to remember the theorem: orthogonal projection flat section cone of revolution onto a plane perpendicular to its axis is a curve of the second order and has one of its foci orthogonal projection to this plane of the vertex of the cone.

Let's consider the construction of section projections when the cutting plane α parallel to one generatrix of the cone (SD).

The cross section will result in a parabola with its vertex at the point A(A`, A"). According to the theorem, the vertex of the cone S projected into focus S`. According to the known =R S` determine the position of the directrix of the parabola. Subsequently, the points of the curve are plotted using the equation p=R.

Constructing section projections when the cutting plane α parallel to one generatrix of the cone, the following can be done:

With the help of auxiliary horizontally projecting planes passing through the top of the cone γ 1 H And γ 2 H.

First they will decide frontal projections points F", G"- at the intersection of the generators S"1", S"2" and the trace of the cutting plane α V. At the intersection of communication lines with γ 1 H And γ 2 H be determined F`, G`.

Other points of the section line can be defined similarly, for example D", E" And D`, E`.

Using auxiliary frontal projection planes ⊥ cone axis γ 3 V And γ 4 V.

Projections of a section of auxiliary planes and a cone onto a plane H, there will be circles. Lines of intersection of auxiliary planes with the cutting plane α there will be front-projecting straight lines.

Gives - a hyperbola when the cutting plane is parallel to the two generators of the cone.

Contents of the article

CONIC SECTIONS, flat curves that are obtained by intersecting a right circular cone with a plane that does not pass through its vertex (Fig. 1). From the point of view of analytical geometry, a conic section is the locus of points satisfying a second-order equation. Except for the degenerate cases discussed in the last section, conic sections are ellipses, hyperbolas, or parabolas.

Conic sections are often found in nature and technology. For example, the orbits of planets revolving around the Sun are shaped like ellipses. A circle is a special case of an ellipse, in which major axis equal to small. A parabolic mirror has the property that all incident rays parallel to its axis converge at one point (focus). This is used in most reflecting telescopes that use parabolic mirrors, as well as in radar antennas and special microphones with parabolic reflectors. A beam emanates from a light source placed at the focus of a parabolic reflector parallel rays. That's why parabolic mirrors are used in high-power spotlights and car headlights. A hyperbole is a graph of many important physical relations, for example, Boyle's law (relating pressure and volume ideal gas) and Ohm's law, which specifies electric current as a function of resistance at constant voltage.

EARLY HISTORY

The discoverer of conic sections is supposedly considered to be Menaechmus (4th century BC), a student of Plato and teacher of Alexander the Great. Menaechmus used a parabola and an equilateral hyperbola to solve the problem of doubling a cube.

Treatises on conic sections written by Aristaeus and Euclid at the end of the 4th century. BC, were lost, but materials from them were included in famous Conic sections Apollonius of Perga (c. 260–170 BC), which have survived to this day. Apollonius abandoned the requirement that the secant plane of the cone's generatrix be perpendicular and, by varying the angle of its inclination, obtained all conic sections from one circular cone, straight or inclined. We are indebted to Apollo and modern names curves - ellipse, parabola and hyperbola.

In his constructions, Apollonius used a two-sheet circular cone (as in Fig. 1), so for the first time it became clear that a hyperbola is a curve with two branches. Since the time of Apollonius, conic sections have been divided into three types depending on the inclination of the cutting plane to the generatrix of the cone. Ellipse (Fig. 1, A) is formed when the cutting plane intersects all generatrices of the cone at the points of one of its cavity; parabola (Fig. 1, b) – when the cutting plane is parallel to one of the tangent planes of the cone; hyperbola (Fig. 1, V) – when the cutting plane intersects both cavities of the cone.

CONSTRUCTION OF CONIC SECTIONS

Studying conic sections as intersections of planes and cones, ancient Greek mathematicians also considered them as trajectories of points on a plane. It was found that an ellipse can be defined as the locus of points, the sum of the distances from which to two given points is constant; parabola - as a locus of points equidistant from a given point and a given straight line; hyperbola - as a locus of points, the difference in distances from which to two given points is constant.

These definitions of conic sections as plane curves also suggest a method for constructing them using a stretched string.

Ellipse.

If the ends of a thread of a given length are fixed at points F 1 and F 2 (Fig. 2), then the curve described by the point of a pencil sliding along a tightly stretched thread has the shape of an ellipse. Points F 1 and F 2 are called the foci of the ellipse, and the segments V 1 V 2 and v 1 v 2 between the points of intersection of the ellipse with the coordinate axes - the major and minor axes. If points F 1 and F 2 coincide, then the ellipse turns into a circle.

Hyperbola.

When constructing a hyperbola, the point P, the point of a pencil, is fixed on a thread that slides freely along pegs installed at points F 1 and F 2 as shown in Fig. 3, A. The distances are chosen so that the segment PF 2 is longer than the segment PF 1 by a fixed value less than the distance F 1 F 2. In this case, one end of the thread passes under the peg F 1 and both ends of the thread pass over the peg F 2. (The point of the pencil should not slide along the thread, so it must be secured by making a small loop on the thread and threading the point through it.) One branch of the hyperbola ( PV 1 Q) we draw, making sure that the thread remains taut at all times, and pulling both ends of the thread down past the point F 2 and when point P will be below the segment F 1 F 2, holding the thread at both ends and carefully etching (i.e. releasing) it. The second branch of the hyperbola ( Pў V 2 Qў ) we draw, having previously swapped the roles of the pegs F 1 and F 2 .

The branches of the hyperbola approach two straight lines that intersect between the branches. These lines, called asymptotes of the hyperbola, are constructed as shown in Fig. 3, b. Angle coefficients these lines are equal ± ( v 1 v 2)/(V 1 V 2), where v 1 v 2 – bisector segment of the angle between asymptotes, perpendicular to the segment F 1 F 2 ; segment v 1 v 2 is called the conjugate axis of the hyperbola, and the segment V 1 V 2 – its transverse axis. Thus, the asymptotes are the diagonals of a rectangle with sides passing through four points v 1 , v 2 , V 1 , V 2 parallel to the axes. To construct this rectangle, you need to specify the location of the points v 1 and v 2. They are at the same distance, equal

from the point of intersection of the axes O. This formula involves the construction of a right triangle with legs Ov 1 and V 2 O and hypotenuse F 2 O.

If the asymptotes of a hyperbola are mutually perpendicular, then the hyperbola is called equilateral. Two hyperbolas that have common asymptotes, but with rearranged transverse and conjugate axes, are called mutually conjugate.

Parabola.

The foci of the ellipse and hyperbola were known to Apollonius, but the focus of the parabola was apparently first established by Pappus (2nd half of the 3rd century), who defined this curve as the locus of points equidistant from a given point (focus) and a given straight line, which is called the director. The construction of a parabola using a stretched thread, based on the definition of Pappus, was proposed by Isidore of Miletus (6th century). Place the ruler so that its edge coincides with the directrix LLў (Fig. 4), and apply the leg to this edge A.C. drawing triangle ABC. We fasten one end of the thread with a length AB at the top B triangle, and the other at the focus of the parabola F. Having pulled the thread with the tip of a pencil, press the tip into variable point P to the free leg AB drawing triangle. As the triangle moves along the ruler, the point P will describe the arc of a parabola with focus F and the headmistress LLў , since total length thread is equal AB, a piece of thread is adjacent to the free leg of the triangle, and therefore the remaining piece of thread PF must be equal to the remaining part of the leg AB, i.e. PA. Intersection point V parabola with an axis is called the vertex of the parabola, the line passing through F And V, – the axis of the parabola. If a straight line is drawn through the focus, perpendicular to the axis, then the segment of this straight line cut off by the parabola is called the focal parameter. For an ellipse and a hyperbola, the focal parameter is determined similarly.

PROPERTIES OF CONIC SECTIONS

Definitions of Pappus.

Establishing the focus of a parabola gave Pappus the idea of giving an alternative definition of conic sections in general. Let F is a given point (focus), and L– a given straight line (directrix) not passing through F, And D F And D L– distance from the moving point P to focus F and headmistresses L respectively. Then, as Pappus showed, conic sections are defined as the locus of points P, for which the relation D F/D L is a non-negative constant. This ratio is called eccentricity e conical section. At e e > 1 – hyperbola; at e= 1 – parabola. If F lies on L, then the geometric loci have the form of straight lines (real or imaginary), which are degenerate conic sections.

The striking symmetry of the ellipse and hyperbola suggests that each of these curves has two directrixes and two foci, and this circumstance led Kepler in 1604 to the idea that the parabola also has a second focus and a second directrix - infinitely remote point and straight. In the same way, a circle can be considered as an ellipse, the foci of which coincide with the center, and the directrixes are at infinity. Eccentricity e in this case is equal to zero.

Dandelen design.

The foci and directrixes of a conic section can be clearly demonstrated by using spheres inscribed in a cone and called Dandelin spheres (balls) in honor of the Belgian mathematician and engineer J. Dandelin (1794–1847), who proposed the following construction. Let a conic section be formed by the intersection of a certain plane p with a two-cavity straight circular cone with apex at a point O. Let us inscribe two spheres into this cone S 1 and S 2 that touch the plane p at points F 1 and F 2 respectively. If the conic section is an ellipse (Fig. 5, A), then both spheres are inside the same cavity: one sphere is located above the plane p, and the other is under it. Each generatrix of the cone touches both spheres, and the locus of the points of contact looks like two circles C 1 and C 2 located in parallel planes p 1 and p 2. Let P – arbitrary point on a conic section. Let's draw straight lines PF 1 , PF 2 and extend the straight line P.O.. These lines are tangent to the spheres at points F 1 , F 2 and R 1 , R 2. Since all tangents drawn to the sphere from one point are equal, then PF 1 = PR 1 and PF 2 = PR 2. Hence, PF 1 + PF 2 = PR 1 + PR 2 = R 1 R 2. Since the planes p 1 and p 2 parallel, segment R 1 R 2 has a constant length. Thus, the value PR 1 + PR 2 is the same for all point positions P, and point P belongs to the geometric locus of points for which the sum of the distances from P to F 1 and F 2 is constant. Therefore, the points F 1 and F 2 – foci of elliptical section. In addition, it can be shown that the straight lines along which the plane p intersects planes p 1 and p 2 , are the directrixes of the constructed ellipse. If p intersects both cavities of the cone (Fig. 5, b), then two Dandelin spheres lie on the same side of the plane p, one sphere in each cavity of the cone. In this case, the difference between PF 1 and PF 2 is constant, and the locus of points P has the shape of a hyperbola with foci F 1 and F 2 and straight lines - intersection lines p With p 1 and p 2 – as headmistresses. If the conic section is a parabola, as shown in Fig. 5, V, then only one Dandelin sphere can be inscribed in the cone.

Other properties.

The properties of conic sections are truly inexhaustible, and any of them can be taken as defining. Important place V Mathematical meeting Pappa (approx. 300), Geometry Descartes (1637) and Beginnings Newton (1687) was occupied with the problem of the geometric location of points relative to four straight lines. If four lines are given on a plane L 1 , L 2 , L 3 and L 4 (two of which may be the same) and a period P is such that the product of the distances from P to L 1 and L 2 is proportional to the product of distances from P to L 3 and L 4, then the locus of points P is a conic section. Mistakenly believing that Apollonius and Pappus were unable to solve the problem of the locus of points relative to four straight lines, Descartes created analytical geometry to obtain a solution and generalize it.

ANALYTICAL APPROACH

Algebraic classification.

In algebraic terms, conic sections can be defined as plane curves whose coordinates in Cartesian system coordinates satisfy the equation of the second degree. In other words, the equation of all conic sections can be written in general view How

where not all coefficients A, B And C are equal to zero. By using parallel transfer and rotation of the axes, equation (1) can be reduced to the form

ax 2 + by 2 + c = 0

px 2 + qy = 0.

The first equation is obtained from equation (1) with B 2 № A.C., the second – at B 2 = A.C.. Conic sections whose equations are reduced to the first form are called central. Conic sections, given by equations second type with q No. 0 are called non-central. Within these two categories there are nine various types conic sections depending on the signs of the coefficients.

2831) if the odds a, b And c have the same sign, then there are no real points whose coordinates would satisfy the equation. Such a conic section is called an imaginary ellipse (or an imaginary circle, if a = b).

2) If a And b have the same sign, and c– opposite, then the conic section is an ellipse (Fig. 1, A); at a = b– circle (Fig. 6, b).

3) If a And b have different signs, then the conic section is a hyperbola (Fig. 1, V).

4) If a And b have different signs and c= 0, then the conic section consists of two intersecting lines (Fig. 6, A).

5) If a And b have the same sign and c= 0, then there is only one real point on the curve that satisfies the equation, and the conic section is two imaginary intersecting lines. In this case we also talk about an ellipse contracted to a point or, if a = b, contracted to a point on the circle (Fig. 6, b).

6) If either a, or b is equal to zero, and the remaining coefficients have different signs, then the conic section consists of two parallel lines.

7) If either a, or b is equal to zero, and the remaining coefficients have the same sign, then there is not a single real point that satisfies the equation. In this case, they say that a conic section consists of two imaginary parallel lines.

8) If c= 0, and either a, or b is also equal to zero, then the conic section consists of two real coincident lines. (The equation does not define any conic section at a = b= 0, because in this case original equation(1) not second degree.)

9) Equations of the second type define parabolas if p And q are different from zero. If p No. 0, a q= 0, we get the curve from step 8. If p= 0, then the equation does not define any conic section, since the original equation (1) is not of the second degree.

Derivation of equations of conic sections.

Any conic section can also be defined as a curve along which a plane intersects a quadratic surface, i.e. with a surface given by a second degree equation f (x, y, z) = 0. Apparently, conic sections were first recognized in this form, and their names ( see below) are due to the fact that they were obtained by intersecting a plane with a cone z 2 = x 2 + y 2. Let ABCD– the base of a right circular cone (Fig. 7) with a right angle at the apex V. Let the plane FDC intersects the generatrix VB at the point F, base – in a straight line CD and the surface of the cone - along the curve DFPC, Where P– any point on the curve. Let's draw through the middle of the segment CD– point E– straight E.F. and diameter AB. Through the point P let's draw a plane, parallel to the base cone intersecting the cone in a circle R.P.S. and direct E.F. at the point Q. Then QF And QP can be taken, accordingly, as the abscissa x and ordinate y points P. The resulting curve will be a parabola.

The construction shown in Fig. 7, can be used for output general equations conic sections. The square of the length of a perpendicular segment restored from any point of the diameter to the intersection with the circle is always equal to the product lengths of diameter segments. That's why

y 2 = RQ H QS.

For a parabola, a segment RQ has a constant length (since at any position of the point P He equal to the segment A.E.), and the length of the segment QS proportional x(from the ratio QS/E.B. = QF/F.E.). It follows that

Where a – constant coefficient. Number a expresses the length of the focal parameter of the parabola.

If the angle at the vertex of the cone is acute, then the segment RQ not equal to the segment A.E.; but the ratio y 2 = RQ H QS is equivalent to an equation of the form

Where a And b– constants, or, after shifting the axes, to the equation

which is the equation of an ellipse. Intersection points of the ellipse with the axis x (x = a And x = –a) and the points of intersection of the ellipse with the axis y (y = b And y = –b) define the major and minor axes, respectively. If the angle at the vertex of the cone is obtuse, then the curve of intersection of the cone and the plane has the form of a hyperbola, and the equation takes the following form:

or, after transferring the axes,

In this case, the points of intersection with the axis x, given by the relation x 2 = a 2, determine the transverse axis, and the points of intersection with the axis y, given by the relation y 2 = –b 2, determine the conjugate axis. If constant a And b in equation (4a) are equal, then the hyperbola is called equilateral. By rotating the axes, its equation is reduced to the form

xy = k.

Now from equations (3), (2) and (4) we can understand the meaning of the names given by Apollonius to the three main conic sections. The terms "ellipse", "parabola" and "hyperbola" come from Greek words, meaning “deficient,” “equal,” and “superior.” From equations (3), (2) and (4) it is clear that for the ellipse y 2 b 2 / a) x, for a parabola y 2 = (a) x and for hyperbole y 2 > (2b 2 /a) x. In each case, the value enclosed in parentheses is equal to the focal parameter of the curve.

Apollonius himself considered only three general type conic sections (types 2, 3 and 9 listed above), but his approach allows a generalization to consider all real second-order curves. If the cutting plane is chosen parallel to the circular base of the cone, then the cross-section will result in a circle. If the cutting plane has only one common point with a cone, its vertex, you get a section of type 5; if it contains a vertex and a tangent to the cone, then we obtain a section of type 8 (Fig. 6, b); if the cutting plane contains two generatrices of the cone, then the section produces a curve of type 4 (Fig. 6, A); when the vertex is transferred to infinity, the cone turns into a cylinder, and if the plane contains two generatrices, then a section of type 6 is obtained.

If you look at a circle from an oblique angle, it looks like an ellipse. The relationship between a circle and an ellipse, known to Archimedes, becomes obvious if the circle X 2 + Y 2 = a 2 using substitution X = x, Y = (a/b) y transform into the ellipse given by equation (3a). Conversion X = x, Y = (ai/b) y, Where i 2 = –1, allows us to write the equation of a circle in the form (4a). This shows that a hyperbola can be viewed as an ellipse with an imaginary minor axis, or, conversely, an ellipse can be viewed as a hyperbola with an imaginary conjugate axis.

Relationship between the ordinates of a circle x 2 + y 2 = a 2 and ellipse ( x 2 /a 2) + (y 2 /b 2) = 1 directly leads to Archimedes' formula A = p ab for the area of the ellipse. Kepler knew the approximate formula p(a + b) for the perimeter of an ellipse close to a circle, but the exact expression was obtained only in the 18th century. after the introduction of elliptic integrals. As Archimedes showed, the area of a parabolic segment is four-thirds the area of an inscribed triangle, but the length of the arc of a parabola could only be calculated after the 17th century. Differential calculus was invented.

PROJECTIVE APPROACH

Projective geometry is closely related to the construction of perspective. If you draw a circle on a transparent sheet of paper and place it under a light source, then this circle will be projected onto the plane below. Moreover, if the light source is located directly above the center of the circle, and the plane and the transparent sheet are parallel, then the projection will also be a circle (Fig. 8). The position of the light source is called the vanishing point. It is indicated by the letter V. If V is not located above the center of the circle or if the plane is not parallel to the sheet of paper, then the projection of the circle takes the shape of an ellipse. With an even greater inclination of the plane, the major axis of the ellipse (projection of the circle) lengthens, and the ellipse gradually turns into a parabola; on a plane parallel to a straight line V.P., the projection has the form of a parabola; with an even greater inclination, the projection takes the form of one of the branches of the hyperbola.

Each point on the original circle corresponds to a certain point on the projection. If the projection has the form of a parabola or hyperbola, then they say that the point corresponding to the point P, is at infinity or infinitely distant.

As we have seen, with a suitable choice of vanishing points, a circle can be projected into ellipses various sizes and with different eccentricities, and the lengths of the major axes do not have direct relationship to the diameter of the projected circle. Therefore, projective geometry does not deal with distances or lengths per se; its task is to study the ratio of lengths that is preserved during projection. This relationship can be found using the following construction. Through any point P plane, draw two tangents to any circle and connect the tangent points with a straight line p. Let another line passing through the point P, intersects the circle at points C 1 and C 2 and straight p- at the point Q(Fig. 9). In planimetry it is proven that PC 1 /PC 2 = –QC 1 /QC 2. (The minus sign arises due to the fact that the direction of the segment QC 1 is opposite to the directions of other segments.) In other words, points P And Q divide the segment C 1 C 2 externally and internally in the same respect; they also say that the harmonic ratio of four segments is equal to - 1. If the circle is projected into a conic section and the same notation is retained for the corresponding points, then the harmonic ratio ( PC 1)(QC 2)/(PC 2)(QC 1) will remain equal to - 1. Point P called a straight pole p relative to the conic section, and the straight line p– polar point P relative to the conic section.

When the point P approaches a conic section, the polar tends to take the position of a tangent; if point P lies on a conic section, then its polar coincides with the tangent to the conic section at the point P. If the point P is located inside the conic section, then its polar can be constructed as follows. Let's draw through the point P any straight line intersecting a conic section at two points; draw tangents to the conic section at the points of intersection; suppose that these tangents intersect at a point P 1. Let's draw through the point P another straight line that intersects the conic section at two other points; let us assume that the tangents to the conic section at these new points intersect at the point P 2 (Fig. 10). Line passing through points P 1 and P 2 , and there is the desired polar p. If the point P approaching the center O central conic section, then polar p moving away from O. When the point P coincides with O, then its polar becomes infinitely distant, or ideal, straight on the plane.

SPECIAL BUILDINGS

Of particular interest to astronomers is the following simple construction of ellipse points using a compass and ruler. Let an arbitrary straight line passing through a point O(Fig. 11, A), intersects at points Q And R two concentric circles centered at a point O and radii b And a, Where b a. Let's draw through the point Q horizontal line, and through R– a vertical line, and denote their intersection point P P when rotating a straight line OQR around the point O there will be an ellipse. Corner f between the straight line OQR and the major axis is called the eccentric angle, and the constructed ellipse is conveniently defined parametric equations x = a cos f, y = b sin f. Excluding the parameter f, we obtain equation (3a).

For a hyperbola, the construction is largely similar. An arbitrary straight line passing through a point O, intersects one of the two circles at a point R(Fig. 11, b). To the point R one circle and to end point S horizontal diameter of another circle, draw tangents intersecting OS at the point T And OR- at the point Q. Let a vertical line passing through a point T, and a horizontal line passing through the point Q, intersect at a point P. Then the locus of the points P when rotating a segment OR around O will be a hyperbola given by parametric equations x = a sec f, y = b tg f, Where f– eccentric angle. These equations were obtained French mathematician A. Legendre (1752–1833). By excluding the parameter f, we get equation (4a).

An ellipse, as N. Copernicus (1473–1543) noted, can be constructed using epicyclic motion. If a circle rolls without slipping along inside another circle with twice the diameter, then each point P, which does not lie on the smaller circle, but is motionless relative to it, will describe an ellipse. If the point P is on a smaller circle, then the trajectory of this point is a degenerate case of an ellipse - the diameter of the larger circle. An even simpler construction of the ellipse was proposed by Proclus in the 5th century. If the ends A And B line segment AB of a given length slide along two fixed intersecting straight lines (for example, along coordinate axes), then each internal point P the segment will describe an ellipse; the Dutch mathematician F. van Schoten (1615–1660) showed that any point in the plane of intersecting lines, fixed relative to a sliding segment, will also describe an ellipse.

B. Pascal (1623–1662) at the age of 16 formulated the now famous Pascal theorem, which states: three points of intersection opposite sides of a hexagon inscribed in any conic section lie on the same straight line. Pascal derived more than 400 corollaries from this theorem.

Municipal Educational Institution

Secondary School No. 4

Conic sections

Completed

Spiridonov Anton

student of class 11A

Checked

Korobeynikova A. T.

Tobolsk - 2006

Introduction

The concept of conic sections

Types of conic sections

Study

Construction of conic sections

Analytical approach

Application

References

Introduction.

Purpose: to study conic sections.

Objectives: learn to distinguish between types of conic sections, construct kinetic sections and apply an analytical approach.

The given proportion can be considered as a system of equations:

This is where the names of curves come from, which were introduced by Apollonius of Perga, who lived in the 3rd century BC: ellipse (έλλείψίς), which means a flaw, a deficiency (of the angle of a cone to a straight line); hyperbola (ύπέρβωλη) - exaggeration, preponderance (of a cone angle over a straight line); parabola (παραβολη) - approximation, equality (of a cone angle to a right angle). Later the Greeks noticed that all three curves could be obtained on one cone by changing the inclination of the cutting plane. In this case, you should take a cone consisting of two cavities and think that they extend to infinity (Fig. 1).

If we draw a section of a circular cone perpendicular to its axis, and then rotate the cutting plane, leaving one point of its intersection with the cone motionless, we will see how the circle will first stretch out, turning into an ellipse. Then the second vertex of the ellipse will go to infinity, and instead of an ellipse you will get a parabola, and then the plane will also intersect the second cavity of the cone and you will get a hyperbola.

The concept of conic sections.

Conic sections are plane curves that are obtained by intersecting a right circular cone with a plane that does not pass through its vertex. From the point of view of analytical geometry, a conic section is the locus of points satisfying a second-order equation. With the exception of degenerate cases discussed in the last section, conic sections are ellipses, hyperbolas or parabolas (Fig. 2).

When a right triangle is rotated about one of its legs, the hypotenuse with its extensions describes a conical surface called the surface of a right circular cone, which can be considered as a continuous series of lines passing through the vertex and called generators, all generators resting on the same circle, called producing. Each of the generators represents the hypotenuse of a rotating triangle (in its known position), extended in both directions to infinity. Thus, each generatrix extends on both sides of the vertex, as a result of which the surface has two cavities: they converge at one point at a common vertex. If such a surface is intersected by a plane, then the section will produce a curve, which is called a conic section. It can be of three types:

1) if a plane intersects a conical surface along all generatrices, then only one cavity is dissected and a closed curve called an ellipse is obtained in the section;

2) if the cutting plane intersects both cavities, then a curve is obtained that has two branches and is called a hyperbola;

3) if the cutting plane is parallel to one of the generatrices, then a parabola is obtained.

If the cutting plane is parallel to the generating circle, then a circle is obtained, which can be considered as a special case of an ellipse. A cutting plane can intersect a conical surface only at one vertex, then the section produces a point, as a special case of an ellipse.

If a plane passing through the vertex intersects both cavities, then the section produces a pair of intersecting lines, considered as a special case of a hyperbola.

If the vertex is infinitely distant, then the conical surface becomes cylindrical, and its section by a plane parallel to the generators gives a pair of parallel lines as a special case of a parabola. Conic sections are expressed by 2nd order equations, the general form of which is

Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0

and are called 2nd order curves.

Types of conic sections.

Conic sections can be of three types:

2) The cutting plane is parallel to one of the tangent planes of the cone; in cross-section, the result is an open curve that goes to infinity - a parabola, lying entirely on one cavity.

Study.

In cases where a conic section has a center of symmetry (center), i.e., is an ellipse or hyperbola, its equation can be reduced (by moving the origin of coordinates to the center) to the form:

a 11 x 2 +2a 12 xy + a 22 y 2 = a 33 .

Further studies of such (called central) conic sections show that their equations can be reduced to an even simpler form:

Ax 2 + Wu 2 = C,

if we choose the main directions for the directions of the coordinate axes - the directions of the main axes (axes of symmetry) of conic sections. If A and B have the same signs (coinciding with the sign of C), then the equation defines an ellipse; if A and B are of different signs, then it is a hyperbole.

CONSTRUCTION OF CONIC SECTIONS.

Studying conic sections as intersections of planes and cones, ancient Greek mathematicians also considered them as trajectories of points on a plane. It was found that an ellipse can be defined as the locus of points, the sum of the distances from which to two given points is constant; parabola - as a locus of points equidistant from a given point and a given straight line; hyperbola - as a locus of points, the difference in distances from which to two given points is constant.

These definitions of conic sections as plane curves also suggest a method for constructing them using a stretched string.

The branches of the hyperbola approach two straight lines that intersect between the branches. These straight lines, called asymptotes of the hyperbola, are constructed as shown in Figure 4, b. Corner

the coefficients of these lines are equal to where is the segment of the bisector of the angle between the asymptotes, perpendicular to the segment F 2 F 1 ; the segment v 1 v 2 is called the conjugate axis of the hyperbola, and the segment V 1 V 2 is its transverse axis. Thus, the asymptotes are the diagonals of a rectangle with sides passing through four points v 1, v 2, V 1, V 2 parallel to the axes. To construct this rectangle, you need to specify the location of points v 1 and v 2. They are at the same distance, equal

from the intersection point of the O axes. This formula assumes the construction of a right triangle with legs Ov 1 and V 2 O and hypotenuse F 2 O.

Parabola. The foci of the ellipse and hyperbola were known to Apollonius, but the focus of the parabola was apparently first established by Pappus (second half of the 3rd century), who defined this curve as the locus of points equidistant from a given point (focus) and a given straight line, which is called headmistress. The construction of a parabola using a tensioned thread, based on the definition of Pappus, was proposed by Isidore of Miletus (VI century) (Fig. 5).

Let's position the ruler so that its edge coincides with the directrix, and apply the drawing leg AC to this edge triangle ABC. Let's fasten one end of the thread of length AB at the vertex B of the triangle, and the other at the focus of the parabola F. Having pulled the thread with the tip of a pencil, press the tip at the variable point P to the free leg AB of the drawing triangle. As the triangle moves along the ruler, point P will describe the arc of a parabola with focus F and directrix, since the total length of the thread is equal to AB, the piece of thread is adjacent to the free leg of the triangle, and therefore the remaining piece of thread PF must be equal to the remaining part leg AB, that is PA. The point of intersection of V of the parabola with the axis is called the vertex of the parabola, the straight line passing through F and V is the axis of the parabola. If a straight line is drawn through the focus, perpendicular to the axis, then the segment of this straight line cut off by the parabola is called the focal parameter. For an ellipse and a hyperbola, the focal parameter is determined similarly.

ANALYTICAL APPROACH

Algebraic classification. In algebraic terms, conic sections can be defined as plane curves whose coordinates in the Cartesian coordinate system satisfy an equation of the second degree. In other words, the equation of all conic sections can be written in general form as

where not all coefficients A, B and C are equal to zero. Using parallel translation and rotation of the axes, equation (1) can be reduced to the form

ax 2 + by 2 + c = 0

The first equation is obtained from equation (1) for B 2 > AC, the second - for B 2 = AC. Conic sections whose equations are reduced to the first form are called central. Conic sections defined by equations of the second type with q > 0 are called non-central. Within these two categories, there are nine different types of conic sections depending on the signs of the coefficients.

1) If the coefficients a, b and c have the same sign, then there are no real points whose coordinates would satisfy the equation. Such a conic section is called an imaginary ellipse (or an imaginary circle if a = b).

2) If a and b have the same sign, and c has the opposite sign, then the conic section is an ellipse; when a = b – circle.

3) If a and b have different signs, then the conic section is a hyperbola.

4) If a and b have different signs and c = 0, then the conic section consists of two intersecting lines.

5) If a and b have the same sign and c = 0, then there is only one real point on the curve that satisfies the equation, and a conic section is two imaginary intersecting lines. In this case, we also speak of an ellipse subtended to a point or, if a = b, a circle subtended to a point.

6) If either a or b is equal to zero, and the other coefficients have different signs, then the conic section consists of two parallel lines.

7) If either a or b is equal to zero, and the remaining coefficients have the same sign, then there is not a single real point that satisfies the equation. In this case, they say that a conic section consists of two imaginary parallel lines.

8) If c = 0, and either a or b is also zero, then the conic section consists of two real coincident lines. (The equation does not define any conic section for a = b = 0, since in this case the original equation (1) is not of the second degree.)

9) Equations of the second type define parabolas if p and q are different from zero. If p > 0 and q = 0, we obtain the curve from step 8. If p = 0, then the equation does not define any conic section, since the original equation (1) is not of the second degree.

Application

Conic sections are often found in nature and technology. For example, the orbits of planets revolving around the Sun are shaped like ellipses. A circle is a special case of an ellipse in which the major axis is equal to the minor axis. A parabolic mirror has the property that all incident rays parallel to its axis converge at one point (focus). This is used in most reflecting telescopes that use parabolic mirrors, as well as in radar antennas and special microphones with parabolic reflectors. A beam of parallel rays emanates from a light source placed at the focus of a parabolic reflector. That's why parabolic mirrors are used in high-power spotlights and car headlights. The hyperbola is a graph of many important physical relationships, such as Boyle's law (relating the pressure and volume of an ideal gas) and Ohm's law, which defines electric current as a function of resistance at a constant voltage

Application

References.

1. Alekseev. Abel's theorem in problems and solutions. 2001

2. Bazylev V. T., Dunichev K. I., Ivanitskaya V. P.. Tutorial for 1st year students of physics and mathematics faculties pedagogical institutes. Moscow "enlightenment" 1974

3. Vereshchagin N.K., A.Shen. Lectures on mathematical logic and theory of algorithms. 1999

4. Gelfand I.M.. Lectures on linear algebra. 1998.

5. Gladky A.V. Introduction to modern logic. 2001

6. M.E. Kazaryan. Course of differential geometry (2001-2002).

7. Prasolov V.V.. Geometry of Lobachevsky 2004

8. Prasolov V.V.. Problems in planimetry 2001

9. Sheinman O.K.. Fundamentals of representation theory. 2004

CONIC SECTIONS, lines that are obtained by cutting a right circular cone with planes that do not pass through its vertex. There are conical cross sections of three types. 1) The cutting plane intersects all generatrices of the cone at points of one of its cavity (Fig., a); the intersection line is a closed oval curve - an ellipse; a circle as a special case of an ellipse is obtained when the cutting plane is perpendicular to the axis of the cone. 2) The cutting plane is parallel to one of the tangent planes of the cone (Fig., b); in cross-section, the result is an open curve that goes to infinity - a parabola, lying entirely on one cavity. 3) The cutting plane intersects both cavities of the cone (Fig., c); the line of intersection - a hyperbola - consists of two identical unclosed parts (branches of the hyperbola) extending to infinity, each of which lies on its own cavity of the cone.

In analytical geometry, conic sections are real, non-disintegrating lines of the second order. In cases where a conic section has a center of symmetry (center), that is, it is an ellipse or hyperbola, its equation in the Cartesian coordinate system can be reduced (by moving the origin of coordinates to the center) to the form a 11 x 2 + 2a 12 xy + a 22 y 2 = a 33, where a 11, a 12, a 22, and 33 are constants. The equations of these curves can be reduced to a simpler form

Aх 2 + Βу 2 = С, (*)

if for the directions of the coordinate axes we choose the so-called main directions - the directions of the main axes (axes of symmetry) of the conic section. If constants A and B have the same signs (coinciding with the sign of C), then equation (*) defines an ellipse; if A and B are of different signs, then it is a hyperbole.

The equation of a parabola cannot be reduced to the form (*). With a proper choice of coordinate axes (one coordinate axis is the only axis of symmetry of the parabola, the other is a straight line perpendicular to it, passing through the vertex of the parabola), its equation can be reduced to the form y 2 = 2 px.

Conic sections were known to mathematicians Ancient Greece. The fact that the ellipse, hyperbola and parabola are sections of cones was discovered by Menaechmus (about 340 BC). Most complete essay, dedicated to these curves, is “Conic Sections” by Apollonius of Perga (about 200 BC). Further development The theory of conic sections is associated with the creation in the 17th century of projective (J. Desargues, B. Pascal) and coordinate (R. Descartes, P. Fermat) methods. With proper choice of the coordinate system (the abscissa axis is the axis of symmetry of the conic section, the ordinate axis is tangent to the vertex of the conic section), the equation of the conic section is reduced to the form y 2 = 2рх + λх 2, where р and λ are constants, р≠0. For λ = 0 this equation defines a parabola, for λ<0 - эллипс, при λ>0 is a hyperbole. This property of the conic section, contained in the last equation, was known to the ancient Greek geometers and was the reason for Apollonius of Perga to assign certain types conic section names that have survived to this day: the word “parabola” means application (since in Greek geometry, the transformation of a rectangle of a given area y 2 into an equal rectangle with a given base 2p is called the application of a given rectangle to this base); the word “ellipse” is a disadvantage (application with a disadvantage); the word “hyperbole” is an excess (an application with excess).

The stereometric definition of a conic section can be replaced by planimetric definitions of these curves as sets of points on a plane. So, for example, an ellipse is a set of points for which the sum of the distances to two given points (foci) has the same value. It is possible to give another planimetric definition of a conic section, covering all three types of these curves: a conic section is a set of points, for each of which the ratio of the distances to a given point (focus) to the distance to a given straight line (directrix) is equal to a given positive number(eccentricity) e. At e<1 коническое сечение - эллипс; при е >1 - hyperbole; when e = 1 - parabola.

Interest in the conic section has always been maintained by the fact that these lines are often found in descriptions various phenomena nature and in human activity. Conic sections acquired special significance after I. Kepler (1609) established through observations, and I. Newton (1687) theoretically substantiated the laws of planetary motion (one of which states that planets and comets solar system move along conic sections, at one of the foci of which the Sun is located).

Lit.: Warden B. L. van der. Awakening Science. 2nd ed. M., 2006; Alexandrov P. S. Lectures on analytical geometry. 2nd ed. M., 2008.