Analytic geometry

MINISTRY OF SCIENCE AND HIGHER EDUCATION OF THE RUSSIAN FEDERATION


           SOUTHERN FEDERAL UNIVERSITY










        Vladimir S. Pilidi





       Analytic Geometry


Textbook










Rostov-on-Don - Taganrog
Southern Federal University Press
2020

UDC 514.122+514.123(075.8)
BBC 22.151.5
  P32


Published by decision of the educational-methodical commission of the I. I. Vorovich Institute of Mathematics, Mechanics, and Computer Science of the Southern Federal University (minutes No. 9 dated September 8, 2020)

Reviewers:
Doctor of physical and mathematical sciences, professor of the Department of Applied Mathematics of the Platov South-Russian State Polytechnic University (NPI), Professor A.E. Pasenchuk;

Candidate of physical and mathematical sciences, associate professor of the Department of Algebra and Discrete Mathematics of the Southern Federal University, Associate professor A.V. Kozak






       Pilidi, V. S.
P32 Analytic geometry : textbook / V.S. Pilidi ; Southern Federal University. - Rostov-on-Don ; Taganrog : Southern Federal University Press, 2020. -195 p.
      ISBN 978-5-9275-3576-7




        The book contains material on analytic geometry included in the university discipline "Algebra and Geometry". In addition to detailed presentation of theoretical material, there are given problems in the volume that is quite sufficient both for practical classes and for students' independent work. Most problems are provided with detailed solutions. The book is addressed to students of the educational program "Theoretical Computer Science and Information Technologies" and can also be used by students of other educational programs.



UDC 514.122+514.123(075.8)
BBC 22.151.5

ISBN 978-5-9275-3576-7


                                               © Southern Federal University, 2020
                                               © Pilidi V.S., 2020

                Preface





    The main method of analytic geometry is the coordinate method. Its essence lies in the fact that geometric objects are associated in some standard way with equations (systems of equations or inequalities). It makes possible to use for research the methods of algebra or mathematical analysis. The concept of the coordinate system, as well as some initial facts from analytic geometry, are already known to students from the school mathematics course.
    The proposed tutorial is based on the author’s lectures on the discipline “Algebra and Geometry” for first-year students of the educational program “Theoretical Computer Science and Information Technologies”. This part of the discipline assumes that, in addition to standard school information, the students know the concept and main properties of determinants, and are familiar with the Gaussian elimination algorithm for solving systems of linear equations.
    The numerous problems given in the book can be the source for practical classes and homework as well as for independent work of students. All these problems are provided with answers. Detailed solutions are given for all problems, save the ones that are solved in one step.
    The sign □ marks the end of the proof and is often omitted when the proof is very short.

Chapter 1





                Straight lines on the plane





        1.1   Coordinates on the plane

    1.1.1 Definition of coordinates

  The place of a point M on a straight line is fully determined by its distance OM from a fixed point O on the line, if we know on which side of the point O is the point M (right or left), see Fig. 1.

-2    —10         1     2_______.    .
         A ’ O B c                      M ”

Fig. 1. Coordinate axis.

It is assumed that on this line there is used some unit in which the distances are measured. The fixed point O is called the origin. The distance OM of the point M 6= O, taken with the sign “plus” if M lies to the right of the origin and with the sign “minus” when M lies to the left from O, is called the coordinate of M.The coordinate of the point O is set to zero. In this case, an arrow is added to the line indicating the positive direction on it.

1.1. Coordinates on the plane

5

    Let us select, on a given line, an arbitrary origin O, a unit of measure, and a definite positive direction. Then any real number, regarded as the coordinate of a point M , fully determines the position of M on this line. And conversely, every point on the line has one and only one coordinate. In general case the coordinate of a point is usually denoted by the letter x, which, as we told above, may be any real number. In this case we write M (x). For instance, A(-2), B(1), C(2) (Fig. 1).
    Such a line is called the coordinate axis. The distance between the points M1 (x1) and M2(x2) on the coordinate axis is found by the formula |M1M2 | = |x2 - x1 |. The midpoint of the segment M1M2 is the point M ⁽xi+x⁾.
    To locate a point on the plane that is, to determine its position, there is used the following well known approach. We suppose that there is defined a unit to measure distances on the plane. We draw two lines at right angles on the plane, the point of intersection of these lines is called the origin and is usually denoted by the letter O. Then we define positive directions on each of the lines denoting these directions by arrows (see Fig. 2, the points on the axes match unit steps).

У N
              B(-3,2)__________









x

O

M

                 C 2, -3)


A(2,3)

D(3, -3)

Fig. 2. Coordinates on the plane

Chapter 1. Straight lines on the plane

We emphasize that on the both lines there is an origin, a unit of measure and a positive direction. Therefore, there are defined coordinates on the both lines. These two lines are called the axes of coordinates. To distinguish these lines one of them is called the axis Ox, x-axis, or axis of abscissas, and the other is called the axis Oy , the y-axis, or axis of ordinates.
    Now we take an arbitrary point A on the plane, and project it on each axis, i.e. we drop the perpendiculars AM and AN from A on the axes. The coordinate x of the point M on the x-axis is called the abscissa of A. The coordinate y of the point N on the y-axis is called the ordinate of A. The position of the point A on the plane is fully determined if its abscissa x and its ordinate y are both given. The two numbers x, y are also called the coordinates of the point A. This fact is denoted as follows: A(x, y). In some cases there is used the notation (x, y) without denoting the point itself. For example, it is possible to say: “let us take the points (1, 2) and (-4, 3)”.
    As a consequence of the correspondence given above, we can state that every point on the plane has two definite real numbers as coordinates; conversely, every pair of real numbers defines one and only one point of the plane.
    The plane equipped with the coordinate system is called the coordinate plane. The coordinate system we are using is called the Cartesian (or rectangular ) coordinate system. The concept of Cartesian coordinates may be generalized to the case when the axes are not necessarily perpendicular to each other, and there are different units along the axes. In the following we don’t use such a generalization.
    The axes divide the plane into four parts which are called the first, second, third and fourth quadrants (Fig. 3). Quadrants are also numbered with Arabic numerals (1, 2, 3, 4) or Roman numerals (I, II, III, IV) and the numbering is counterclockwise. The inequalities defining the quadrants are given below.

Quadrant I: x > 0, y > 0;

quadrant II: x < 0, y > 0;

1.1. Coordinates on the plane

7

quadrant III: x < 0, y < 0; quadrant IV: x > 0, y < 0.

У


Second quadrant

First quadrant

Third               Fourth

quadrant

quadrant

O

Fig. 3. Quadrants on the plane


    1.1.2  Distance between two points


   Let us consider two points M1(x1, y1) and M2(x2, y2) on the plane. To find the distance d between these points, we drop perpendiculars M₁P₁ , M2P2 on the x-axis and perpendiculars M1Q1 , M2Q2 on the y-axis (Fig. 4).
   We get the right triangle AM1M2R with the legs M1R, M2R and the hypotenuse M1M2. From the equalities
M1R=P1P2=      |x2 - x1|, M2R = Q1Q2 = |y2 - y1|,
and the Pythagorean theorem we get that
d² = (x2 - x1)² + (y2 - y1)², d = \ -x - xi)² + (У2 — У1)² .

    The distance d between the points M1(x1, y1) and M2(x2, y2) is obtained by the formula d = (x(x2 — xi)² + (y2 — yi )²'.

Chapter 1. Straight lines on the plane


    1.1.3 Midpoint of a Segment


    Let us take two points A(a1, a2) and C (c1, c2) on the coordinate plane. Coordinates of the midpoint B of the segment AC are the arithmetic means of the corresponding coordinates of A and C, that is, if B(b₁, b₂) then


a1 + c1   a2 + c2
b¹ = —T~, b² = ^~


To prove the first relation we drop the perpendiculars AP , BQ and CR on the x-axis (see Fig. 5). Then the point Q becomes the midpoint of the segment P R. On the x-axis the points P and R have the following coordinates: P (a1) and R(c1). Therefore the point Q has the coordinate a1+ c¹, and this value is the first coordinate of the point B. Similarly we get that the second coordinate of the point B is a²^², so B (a+c¹, a²+²).



   The midpoint of the segment AC where A(a1, a2), C (c1, c2) is the point B (', a²++c²).

1.1. Coordinates on the plane

9


    1.1.4  Vectors on the coordinate plane



    A vector is a directed segment of a line, that is, a segment for which it is indicated which of its boundary points is the beginning point and which is the ending point. The vector with the beginning point A and the ending point B is usually denoted by aB. The points A and B are also called the tail and head of the vector ABB, respectively. Vectors are sometimes indicated by small Latin letters with the arrow (sometimes with the dash) above them, for example ~a or a0. Another common use is to mark vectors in bold characters: a. The length |AB | of the vector AB is defined by the equality |ABB | = |AB |, i.e. it is the length of the segment AB . In the special case when the points A and B coincide then the vector AB = AA is





denoted by 0 (or 0 and 0) and is called the zero vector. The length of the zero vector equals zero and its direction is assumed to be undetermined. It is assumed that the vector may be displaced parallel to itself. It means that if the vectors AB and A₁B₁ have equal lengths, i.e. |AB| = |A₁B₁ | and the same directions they are considered to be equal (see Fig. 6).

Chapter 1. Straight lines on the plane

TV <• -n 1         j    -» Г*    -»
Fig. 6.  Equal vectors: ~a = b = ~c



   Vectors lying on parallel straight lines are called collinear ¹⁾. The zero vector is considered to be collinear with any vector.

Fig. 7. Collinear vectors



    Remark. It should be noted that the binary relation “two vectors are collinear” in the case of vectors on the plane (and in the case of vectors in the space considered below) is not transitive. It means that collinearity of vectors ~a and ~b and collinearity of vectors ~b and ~c do not necessarily


   ¹⁾ We use the definition of parallel lines, which is slightly different from the “school” one. We assume that two straight lines are parallel if they are equal, or if they have no common points. Such a definition is also used in mathematical texts. In our case it somewhat facilitates the statements about collinear vectors and parallel straight lines.