Guess the meaning of the following words.

fundamental (a) ["fAndq'mentl] object (n) ['ObGIkt]

interpretation (n) [Intq:prI'teISqn] unique (a) [ju: 'nI:k]

remark (n) [rI'ma:k] analytic (a) ["xnq'lItIk]

problem (n) ['prOblqm] figure (n) ['fIgq]

converse (a) ['kOnvq:s] pair (n) [pFq]

coordinate (n) [koq'O:dnIt]} nature (n) ['neItSq]

 

Read and learn the following words:

preliminary (a) [prI'lImInqrI] предварительный

determine (v) [dI'tq:mIn] определять

essentially (adv) [I'senSqlI] зд. по существу

essential (a) [I'senSql] существенный

strictly (adv) ['strIktlI] строго, точно

closely (adv) ['klquslI] зд. тесно, близко

to constitute (v) ['kOnstItju:t] составлять

characteristic (n) ["kxrIktq'rIstIk] свойство, особенность

further (a) ['fq:Dq] другой, дальнейший

initially (adv) [I'nISqlI] вначале, с самого начала

rather (adv) ['ra:Dq] скорее

convenience (n) [kqn'vi:njqns] удобство

convention (v) [kqn'venSqn] условие

to emphasize (v) ['emfqsaIz] подчеркивать

strongly (adv) ['strONlI] сильно, строго

to locate (v) [lqu'keIt] определить местоположение

implication (n) ["ImplI'keISen] смысл, значение

to imply (v) [Im'plaI]{ значить, означать, подразумевать

carefully (adv) ['kFqfqlI] осторожно

to note (v) [nout] отмечать, обращать внимание

familiar (a) [fq'mIljq] хорошо известный, знакомый

accordingly (adv) [1q'kO:dINlI] таким образом, поэтому

path (n) [pa:T] траектория

to trace (v) [treIs] чертить, начертать

necessarily (adv) ['nesIsqrIlI] обязательно, непременно

to summarize (v) ['sAmqraIz] суммировать, подводить итог

to enable (v) [I'neIbl] давать возможность, право

NOTES:

 

to focus attention сосредоточить внимание

totality of points совокупность точек

Reading Activity

TEXT A

EQUATION AND LOCUS

 

Two fundamental problems of analytic geometry. In this chapter we shall make a preliminary study of the following two fundamental problems of analytic geometry:

I. Given an equation, to determine its geometric interpretation or representation.

II. Given a geometric figure or condition, to determine its equation or analytic representation.

The students will note that these problems are essentially converses of each other. Strictly speaking, however, both problems are so closely related that together they constitute the fundamental problem of all analytic geometry. For example, we shall see later that, after obtaining the equation for a given geometric condition, it is often possible by a study of this equation to determine further geometric characteristics and properties for the given condition. Our purpose in initially considering two separate problems is not one of necessity, but rather one of convenience; we are thus enabled to focus our attention on fewer ideas at a time.

First Fundamental Problem. The Locus of an Equation.

Assume that we are given an equation in the two variables x and y, which we may write briefly in the form

f (x, y) = 0(1)

In general there are infinitely many pairs of values of x and y which satisfy this equation. Each such pair of real values will be taken as the coordinates

(x, y) of a point in the plane.

This convention is the basis of Definition 1. The totality of points, and only those points, whose coordinates satisfy an equation (1), is called the locus or graph of the equation.

Another convenient expression is given by Definition 2. Any point whose coordinates satisfy an equation (1) is said to lie on the locus of the equation.

It cannot be emphasized too strongly that only those points whose coordinates satisfy an equation lie on its locus. That is, if the coordinates of a point satisfy an equation, that point lies on the locus of the equation; and conversely, if a point lies on the locus of an equation, its coordinates satisfy the equation. Since the coordinates of the point of a locus are restricted by its equation, such points will in general be located in positions which, taken together, form a definite path called a curve as well as a graph or locus.

Second fundamental problem. We will now consider the second fundamental problem of analytic geometry.

A geometric figure, such as a curve, is generally given by its definition. By the definition of an object is meant a description of that object of such a nature that it is possible to identify it definitely among all other objects of its class. The implication of this statement should be carefully noted: it expresses a necessary and sufficient condition for the existence of the object defined. Thus, let us consider that we are defining a plane curve of type C by means of a unique property P which C possesses. Then, in the entire class of all plane curves, a curve is of type C if and only if it possesses property P.

As a specific example, let us consider that familiar plane curve, the circle. We define a circle as a plane curve possessing the unique property P that all its points are equally distant from a fixed point in its plane. This means that every circle has property P; and conversely, every plane curve having property P is a circle.

For a curve, a geometric condition is a law which the curve must obey. This means that every point on the curve must satisfy the particular law for the curve. Accordingly a curve is often defined as the locus or path traced by a point moving in accordance with a specified law. Thus, a circle may be defined as the locus moving in a plane so that it is always at a constant distance from a fixed point in that plane. A locus need not necessarily satisfy a single condition; it may satisfy two or more conditions. Thus, we may have a curve which is the locus of a point moving so that it passes through a given point, and it is always at a constant distance from a given line. We may then summarize the preceding remarks in the followingdefinition:

A curve is the locus of all those points, and only those points, which satisfy one or more given geometric conditions.

The student should note that this definition implies that the given condition or conditions are both necessary and sufficient for the existence of the curve.

Post-Reading Activity.

Ex. 8. Answer the following questions:

1. What are the two fundamental problems of analytic geometry? 2. How many pairs of values of x and y satisfy the equation F(x, y) = 0? 3. What is the locus or graph of the equation? 4. What point lies on the locus of the equation F(x, y) = 0? 5. What are the coordinates of the point of a locus restricted by? 6. By what is a geometric figure generally given? 7. How may a circle be defined? 8. How many conditions may a locus satisfy? 9. What is the difference between the first and the second problems?

Ex. 9. Match the English words and word combinations with their Russian equivalents.

1. a familiar plane curve	a. совокупность точек
2. to lie on the locus	b. весь класс
3. strictly speaking	c. как необходимые, так и достаточные условия
4. two fundamental problems	d. известная плоская кривая
5. to write briefly	e. обязательно удовлетворять
6. the totality of points	f. точно говоря
7. the implication of a statement	g. единственное условие
8. both necessary and sufficient conditions	h. предварительное изучение
9. a preliminary study	i. тесно связанный
10. necessarily satisfy	j. находиться на графике
11. the entire class	k. записывать кратко
12. a single condition	l. две основные задачи
13. to pass through a given point	m. проходить через данную точку
14. closely related	n. смысл утверждения

 

Ex. 10. Give the corresponding plural forms of the following nouns.

a) us [əs] → i [aI] focus → foci

Calculus, genius, locus, modulus, nucleus, radius.

b) is [Is] → es [I:z] axis → axes

Thesis, emphasis, analysis, basis, hypothesis, crisis, phasis, parenthesis.

c) ix [Iks] → es [I:z] matrix → matrices

ex [əks] → es [I:z] vertex → vertices

Directrix, bisectrix, index.

d) on [ ɔn ] → a[ə] polyhedron → polyhedra

um [əm] → a [ə] datum → data

Continuum, medium, spectrum, minimum, maximum, phenomenon, criterion.

e) a [ə] → ae [i:] formula → formulae (formulas)

Abscissa, hyperbola, lacuna, corona.

 

Ex. 11. Use the plural and singular forms of the nouns given in Ex.10. The first letters of the words are given.

 

1. The area of an ellipse equals π times the product of the long and the short r … 2. If a curve is symmetric with respect to both a …, is it symmetric with respect to the origin? 3. Analytic methods give us a means of finding the equations of l … 4. The notion of a four-dimensional geometry is a very helpful one in studying physical p … 5. Find the equation of the ellipse with f … at the points (0, 4). 6. In each of the following h …, locate the vertices and f … 7. All these facts may serve as reference d … 8. C … is a branch of mathematics divided into two parts differential calculus and mathematical calculus. 9. Circular area is measured by its r … 10. Einstein was a mathematical g …

 

Ex. 12. Ask disjunctive questions (tag-questions).

 

1. To draw the graph of a function isn’t difficult,...? 2. Don’t try to obtain the equation of a locus quickly,...? 3. Every student must understand the implication of the equation of a locus,...? 4. Let’s do a preliminary study of the function,...? 5. This theorem constitutes a common property of fields,...? 6. For convenience we had to focus attention on the characteristic of the locus,...? 7. They have studied the entire class of plane curves,...? 8. You should know the coordinates of a point to determine its position in a plane,...? 9. There is an analytic interpretation of the equation of a locus,...? 10. I’m to study the course of analytic geometry,...?

 

Ex. 13. Find out whether the statements are true or false. Use introductory phrases.

 

Exactly. Quite so. I fully agree to it. I don’t think this is the case.

Quite the contrary. Not quite. It’s unlikely. Just the reverse.

 

1. There exists a close relationship between two fundamental problems of analytic geometry.

2. After obtaining the equation for a given geometric condition it is impossible to determine further geometric properties for the given condition.

3. There is only one pair of values x and y satisfying the equation F(x, y)=0.

4. The totality of points satisfying Equation (1) is called the locus or graph of the equation.

5. Any point whose coordinates satisfy Equation (1) is said not to lie on the locus of the equation.

6. A geometric figure, such as a curve is generally given by its formula.

7. A circle possesses a unique property that all its points are equally distant from the points in its plane.

8. For the curve, a geometric condition is a law which the curve must obey.

9. A curve is the locus of the points which satisfy one and only one condition.

10. There is some difference between the first and the second problems of analytic geometry.

 

Ex. 14. Say these sentences in English.

1. Нам предстоит рассмотреть методы анализа линейных уравнений.

2. Во многих текстах чертёж уравнения называется кривой, даже если это прямая линия.

3. Такая кривая называется геометрическим местом точек уравнения.

4. Геометрическим местом точек уравнения (его графиком) является кривая, содержащая точки и только те точки, координаты которых удовлетворяют этому уравнению.

5. Иногда кривая может быть определена множеством условий, а не уравнением, хотя уравнение может быть получено из данных условий.

6. В этом случае рассматриваемая кривая являлась бы графиком всех точек на плоскости, которые соответствовали бы этим уравнениям.

7. Например, можно сказать, что кривая – это геометрическое место всех точек на плоскости, расположенных на фиксированном расстоянии от одной фиксированной точки, называемой центром окружности.

8. Прямая линия может быть определена как геометрическое место всех точек на плоскости, равноудалённых от двух фиксированных точек.

9. Метод выражения множества условий в аналитической форме даёт уравнения.

10. Основные понятия геометрического места точки в геометрии тесно связаны с понятием уравнения в алгебре.

 

 

Ex.15. Read and translate the following sentences. Group them according to the models.

Models. a) He should help us. (obligation). Он должен нам помочь.

 

b) You should have helped us. Вам следовало бы нам помочь.

(negative probabilities; unwanted things).

 

c) We answered that we should help him. Мы ответили, что поможем ему.

(reported speech; expresses future action).

 

1. I should have indicated the directions in which the distances were to be measured. 2. We answered that we should explain the methods by which the results had been obtained. 3. The professor said we should determine the equation of that geometric figure. 4. I replied that I should have written the thesis by the end of the year. 5. A circle is a plane curve which should satisfy a unique property. 6. This locus should be investigated by the students. 7. You should have solved more difficult puzzles, because your intelligence is above average. 8. Students should know more about transcendental functions. 9. Any point on the curve should possess the unique property.

 

Ex. 16. Topics for discussion.

1. Dwell on the fundamental problems of all analytic geometry.

2. Speak on the locus of an equation.

3. Describe the second fundamental problem of analytic geometry.

Ex. 17. Read the text and answer the following questions.

 

1. What is the parabola? (The ellipse, the hyperbola).

2. What does the equation of the parabola depend on?

3. How do we call the points in which the ellipse cuts the principal axis?

4. Can we consider the notion of correspondence between a geometric locus and an equation as a general concept?

5. Is there any relationship between the parabola, the ellipse and the hyperbola?

 

Text B

PARTICULAR SPECIES OF LOCI

 

We shall proceed to the discussion of particular species of loci, namely, a parabola, an ellipse and a hyperbola.

The parabola is the locus of points which are equidistant from a fixed point and a fixed straight line.

The fixed point is the locus, the fixed line is the directrix. The line perpendicular to the directrix and passing through the focus is the axis of the parabola. The axis of the parabola is, obviously, a line of symmetry. The point on the axis halfway between the focus and the directrix on the parabola is the vertex of the parabola. The parabola is fixed when the focus and the directrix are fixed. The equation of the parabola, however, depends on the choice of the coordinate system. If the vertex of the parabola is at the origin and the focus is at the point (О, Р), its equation is X² = 2PY or Y² =2PX.

The ellipse is the locus of a point which moves so that the sum of its distances from two fixed points called the foci is constant. This constant will be denoted by 2a, which is necessarily greater than the distance between the foci (the focal distance). The line through the foci is the principal axis of the ellipse; the points in which the ellipse cuts the principal axis are called the vertices of the ellipse. If the centre of the ellipse is at the origin but the foci are on the y-axis its equation is

+ = 1

where a and b represent the lengths of its semimajor and semiminor axes (большая и малая полуоси).

The hyperbola is the locus of a point which moves so that the difference of its distances from two fixed points is a constant 2a. Its equation is

- = 1

This equation shows that the hyperbola is symmetric with respect to both coordinate axes and also the origin. It intersects the X -axis but does not cut the Y -axis. Hence, the curve is not contained in a bounded portion of a plane. The curve consists of two branches. The line segment joining the vertices is called the transverse axis of the hyperbola; its length is 2a. The point midway between the vertices is a geometrical centre and is called the centre of the hyperbola.

 

UNIT V