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Unformatted text preview: Math 454, Spring 08, homework assignment #1, due Jan 31 Problem 1 Let OA = 2 a be the diameter of a circle (where O is the origin) and OY and AV be the tangents to the circle at 0 and A , respectively. A half-line l is drawn from O and meets the circle at C and the line AV at B . On OB mark off the segment OP = CB . If we rotate l about O , the point P will describe a curve known as the cissoid of Diocles . By taking OA as the x-axis and OY as the y-axis, prove the following statements. 1. The cissoid can be given by the parametric equation ( x, y ) = r ( t ) = 2 at 2 1 + t 2 , 2 at 3 1 + t 2 , t R . (What is the meaning of the parameter t ?) 2. The origin is the singular point of the cissoid. 3. As t , r ( t ) approaches the line x = 2 a and r ( t ) (0 , 2 a ) (in other words, x = 2 a is an asymptote of the cissoid). 4. What is the connection between this definition of the cissoid and Exercise 1.15 in Pressley?...
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