p0156

# p0156 - a 2 b 2 to both sides yields x 2 − 2 bx b 2 y 2...

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1.56. CHAPTER 1, PROBLEM 56 63 1.56 Chapter 1, Problem 56 Problem: Show that r =2 a sin θ +2 b cos θ defines a circle. Determine the circle’s radius, R ,andthe location of its center in both cylindrical and Cartesian coordinates. Solution: Our first step is to rewrite the equation in Cartesian coordinates. Thus, we begin by multiplying both sides by r , i.e., r 2 =2 ar sin θ +2 br cos θ Now, we know that r 2 = x 2 + y 2 , r sin θ = y and r cos θ = x . Therefore, x 2 + y 2 =2 ay +2 bx = x 2 2 bx + y 2 2 ay =0 Finally, adding
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Unformatted text preview: a 2 + b 2 to both sides yields x 2 − 2 bx + b 2 + y 2 − 2 ay + a 2 = a 2 + b 2 = ⇒ ( x − b ) 2 + ( y − a ) 2 = a 2 + b 2 This is the equation for a circle of radius √ a 2 + b 2 centered at x = b , y = a . In terms of cylindrical coordinates, we know that r = x 2 + y 2 and θ = tan − 1 ( y/x ) . Hence, the circle’s center is located at r = a 2 + b 2 , θ = tan − 1 p a b Q...
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