# A Collection of Problems in Analytical Geometry

### Analytical Geometry in the Plane

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A Collection of Problems in Analytical Geometry, Part I: Analytical Geometry in the Plane is a collection of problems dealing with higher analytical geometry. The book discusses elementary problems dealing with plane analytical geometry. The text presents topics on the axis and intervals on an axis and coordinates on a straight line. The book also defines what a rectangular Cartesian coordinates in a plane is, the division of an interval in a given ratio, and shows how to calculate the area of a triangle. The equation of a curve, the functions of two variables, and the concept of an equation of a curve are explained by the use of examples and problems. The author also addresses the geometrical properties of curves of the second order, the equations of a straight line, a circle, an ellipse, a hyperbola, and a parabola. The text then discusses the general theory of second-order curves and emphasizes the equations of the central curves of the second order. The author cites the simplification of these equations as being applicable to theoretical mechanics. This collection of problems can be used by teachers of analytical geometry and their students.

## Table of Contents

Author's Preface to the First Russian Edition

Author's Preface to the Second Russian Edition

Part I. Analytical Geometry in the Plane

1. Elementary Problems in Plane Analytical Geometry

§ 1. The Axis and Intervals on an Axis. Coordinates on a Straight Line

§ 2. Rectangular Cartesian Coordinates in a Plane

§ 3. Polar Coordinates

§ 4. Directed Intervals. Projection of an Interval on to an Arbitrary Axis. Projection of an Interval on to a Coordinate Axis. Length and Polar Angle of an Interval. Distance between Two Points

§ 5. Division of an Interval in a Given Ratio

§ 6. Area of a Triangle

§ 7. Transformation of Coordinates

2. Equation of a Curve

§ 8. Functions of Two Variables

§ 9. The Concept of an Equation of a Curve. The Specification of a Curve by Means of an Equation

§ 10. Derivation of Equations for Previously Defined Curves

§ 11. Parametric Equations of a Curve

3. Curves of the First Order

§ 12. General Equation of a Straight Line. Equation of a Straight Line in Terms of Its Gradient. Angle between Two Straight Lines. Condition for Two Lines to be Parallel or to be Perpendicular

§ 13. Degenerate Equations of a Straight Line. Equation of a Straight Line in Terms of Its Intercepts on the Axes. Compatibility of the Equations of Two and Three Lines

§ 14. The Normal Equation of a Straight Line. How to Calculate the Distance of a Point from a Straight Line

§ 15. Equation of a Pencil of Straight Lines

§ 16. The Polar Equation of a Straight Line

4. Geometrical Properties of Curves of the Second Order

§ 17. The Circle

§ 18. The Ellipse

§ 19. The Hyperbola

§20. The Parabola

§ 21. The Polar Equation of the Ellipse, Hyperbola, and Parabola

§ 22. Diameters of Curves of the Second Order

5. Simplification of the General Equation of a Curve of the Second Order. The Equations of Certain Curves Commonly Encountered in Mathematics and Allied Subjects

§ 23. The Center of a Curve of Second Order

§ 24. Reduction of the Equation of a Central Curve of the Second Order to Its Simplest Form

§ 25. Reduction of a Parabolic Equation to Its Simplest Form

§ 26. The Equations of Certain Curves Commonly Encountered in Mathematics and Allied Subjects

Answers to Questions and Notes on Solutions

## Product details

- No. of pages: 196
- Language: English
- Copyright: © Pergamon 1966
- Published: January 1, 1966
- Imprint: Pergamon
- eBook ISBN: 9781483155845

## About the Author

### D. V. Kletenik

## About the Editors

### W. J. Langford

### E. A. Maxwell

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