Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 192.2 Polar CoordinatesPolar Coordinate SystemsA polar coordinate system in a plane consists of a fixed point O, called the pole (ororigin), and a ray emanating from a pole, called the polar axis. In such a coordinatesystem we can associate with each point P in the plane a pair of polar coordinates (r,θ),where r is the distance from P to the pole and θ is an angle from the polar axis to the rayOP.P(r,θ)rθO OriginPolar axisThe number r is called the radial distance of P and the number θ the angular coordinate(or polar angle) of P.Example 2.9 Plot the points whose polar coordinates are given.(a) (3,π/4) (b) (4,2π/3) (c) (2,5π/4) (d) (4,11π/6)Solution .........The polar coordinates of a point are not unique. For example, the polar coordinatesall represent the same point.5π/3(1, 5π 3 ) −π/3(1,5π/3), (1,−π/3), (1,11π/3)(1,− π 3 ) 11π/3In general, if a point P has polar coordinates (r,θ), then(r,θ +2nπ) and(r,θ −2nπ),(1, 11π3 )are also polar coordinates of P for any nonnegative integer n.As define above, the radial coordinate r of a point P is nonnegative, since it representsthe distance from P to the pole. However, it will be convenient to allow for negativevalues of r as well. To motivate an appropriate definition, consider the point P with polarcoordinates (3,5π/4). We can reach this point by