Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 24Families of CirclesWe will consider three families of circles in which a is assumed to be a positive constant:r = a r = 2acosθ r = 2asin2θ• The equation r = a represents a circle of radius a, centered at the pole• The equation r = 2acosθ represents a circle of radius a, centered on the x-axis andtangent to the y-axis at the origin.• The equation r = a represents a circle of radius a, centered on the y-axis and tangentto the x-axis at the origin.π/2π/2π/2r = 2asinθar = a0r = −2acosθ r = 2acosθ•(a,π)•(a,0)0•(a, π 2 )0•(a,− π 2 )r = −2asinθExample 2.16 Sketch the graphs of the following equations in polar coordinates.(a) r = 4cosθ (b) r = −5sinθ (c) r = 3Solution .........Families of Rose CurvesIn polar coordinates, equations of the formr = asinnθ or r = acosnθin which a > 0 and n is a positive integer represent families of flower-shaped curves calledroses. The rose consists of n equally spaced petals of radius a if n is odd and 2n equallyspaced petals of radius a if n is even.