Analytic Geometry (292,00 руб.)

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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

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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

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Contents 1 Straight lines on the plane 4 1.1 Coordinates on the plane . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Definition of coordinates . . . . . . . . . . . . . . . . 4 1.1.2 Distance between two points . . . . . . . . . . . . . . 7 1.1.3 Midpoint of a Segment . . . . . . . . . . . . . . . . . 8 1.1.4 Vectors on the coordinate plane . . . . . . . . . . . . 9 1.2 Straight line on the plane . . . . . . . . . . . . . . . . . . . 17 1.2.1 Equations of straight lines . . . . . . . . . . . . . . . 17 1.2.2 Special cases of equations of straight lines . . . . . . 20 1.2.3 Distance of a point from a line . . . . . . . . . . . . . 21 1.2.4 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . 24 1.2.5 Pencils of straight lines . . . . . . . . . . . . . . . . . 27 1.2.6 Normal equations of straight lines . . . . . . . . . . . 29 1.3 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4 Some tasks on straight lines on the plane . . . . . . . . . . . 34 2 Second order curves on the plane 37 2.1 The circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 The ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 192

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Contents 193 2.2.1 Definition and equation of the ellipse . . . . . . . . . 43 2.2.2 Directrices . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2.3 Tangent to the ellipse . . . . . . . . . . . . . . . . . . 50 2.2.4 Parametric equations of the ellipse . . . . . . . . . . 53 2.3 The hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.3.1 Definition and equation of the hyperbola . . . . . . . 54 2.3.2 Directrices . . . . . . . . . . . . . . . . . . . . . . . . 60 2.3.3 Tangent to the hyperbola . . . . . . . . . . . . . . . 62 2.4 The parabola . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.5 Polar equations of second order curves . . . . . . . . . . . . 68 3 Analytic geometry in the space 70 3.1 Cartesian coordinates in the space . . . . . . . . . . . . . . . 70 3.1.1 Distance between two points . . . . . . . . . . . . . . 72 3.1.2 Vectors in the space . . . . . . . . . . . . . . . . . . 73 3.2 Plane in the space . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2.1 Equation of the plane . . . . . . . . . . . . . . . . . . 86 3.2.2 Special cases of equations of planes . . . . . . . . . . 89 3.2.3 Distance of a point from a plane . . . . . . . . . . . . 91 3.2.4 Equation of the plane in the intercept form . . . . . . 94 3.2.5 Pencils and bundles of planes . . . . . . . . . . . . . 95 3.2.6 Normal equation of the plane . . . . . . . . . . . . . 98 3.3 Straight line in the space . . . . . . . . . . . . . . . . . . . . 99 3.4 Some tasks on straight lines and planes in the space . . . . . 107 4 Problems 110

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194 Contents 4.1 Straight lines on the plane . . . . . . . . . . . . . . . . . . . 110 4.2 Second order curves on the plane . . . . . . . . . . . . . . . 114 4.3 Analytic geometry in the space . . . . . . . . . . . . . . . . 124 5 Answers and solutions 134 5.1 Straight lines on the plane . . . . . . . . . . . . . . . . . . . 134 5.2 Second order curves on the plane . . . . . . . . . . . . . . . 141 5.3 Analytic geometry in the space . . . . . . . . . . . . . . . . 164

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