Chapter 1 Topics in Analytic Geometry
Chapter 1 Topics in Analytic Geometry 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.
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 25r = asin2θ r = asin3θ r = asin4θr = asin5θr = asin6θr = acos2θ r = acos3θ r = acos4θr = acos5θr = acos6θFamilies of Cardioids and LimaconsEquations with any of the four formsr = a±bsinθr = a±bcosθin which a > 0 and b > 0 represent polar curves called limacons. There are four possibleshapes for a limacon that can be determined from the ratio a/b. If a = b (the case a/b = 1),then the limacons is called a cardioids because of its heart-shaped appearance.
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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 25r = as<strong>in</strong>2θ r = as<strong>in</strong>3θ r = as<strong>in</strong>4θr = as<strong>in</strong>5θr = as<strong>in</strong>6θr = acos2θ r = acos3θ r = acos4θr = acos5θr = acos6θFamilies of Cardioids and LimaconsEquations with any of the four formsr = a±bs<strong>in</strong>θr = a±bcosθ<strong>in</strong> which a > 0 and b > 0 represent polar curves called limacons. There are four possibleshapes for a limacon that can be determ<strong>in</strong>ed from the ratio a/b. If a = b (the case a/b = 1),then the limacons is called a cardioids because of its heart-shaped appearance.
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