Introduction to Analytic Geometry Chapter No. 4
Geometry, one of the oldest branches of mathematics, was systematically studied by the Greeks around four centuries B.C. Euclid, in 300 B.C., played a significant role in shaping the geometry taught in schools through his thirteen books on the subject. Later, in the 17th century, French philosopher and mathematician René Descartes introduced algebraic methods in geometry, leading to the birth of analytic geometry, also known as coordinate geometry. This book aims to present the fundamentals of this subject.
Coordinate System
To establish a coordinate system, draw two mutually perpendicular number lines in a plane, labeled x’x and y’y. These lines intersect at a point called the origin (O). The horizontal line x’Ox is the x-axis, and the vertical line y’Oy is the y-axis. The convention is that points above x’Ox on the y-axis correspond to positive real numbers, while those below correspond to negative real numbers. Similarly, points to the right of O on the x-axis are positive, and those to the left are negative.
Locating Points in the Plane
Any point (P) in the plane can be located using an ordered pair of real numbers, denoted as (x, y). Drawing lines parallel to the coordinate axes through P and meeting the x-axis at R and y-axis at S, the directed distance OR is x, and the directed distance OS is y. Thus, the ordered pair (x, y) provides the necessary information to locate point P. Conversely, any ordered pair (x, y) can be associated with exactly one point in the plane. This method of pairing points with ordered pairs is known as the two-dimensional rectangular (or Cartesian) coordinate system.
Exercise 4.1
Exercise 4.2
Exercise 4.3
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Exercise 4.4
Exercise 4.5