Find parametric equations for the path a particle that moves along the circle x2+(y−1)2=4 Find parametric equations for the path a particle that moves along the circle x2+(y−1)2=4. In the manner describe a) One around clockwise starting at (2,1) b) Three times around counterclockwise starting at (2,1) c) halfway around counterclockwise starting at (0,3)

Question

Find parametric equations for the path a particle that moves along the circle x2+(y−1)2=4  Find parametric equations for the path a particle that moves along the circle  x2+(y−1)2=4.  In the manner describe  a) One around clockwise starting at (2,1)  b) Three times around counterclockwise starting at (2,1)  c) halfway around counterclockwise starting at (0,3)

Joseph Fair · Accepted Answer

HINT : Compare  (x2)+(y−12)2=1  with cos2t+sin2t=1.  EDIT:  Drawing the circle will help. In your case, since one around clockwise, y-coordinate of the point has to be decreasing first. This is the reason why the sint is with minus. Setting t=0,π6,π4,… will also help.

alkaholikd9 · Accepted Answer

Step 1  A particle that moves along the circle x2+(y−1)2=4.  Step 2  (a) The path of a particle that moves along the circle x2+(y−1)2=4 in the manner once around clockwise, starting at (2,1).  x2+(y−1)2=4  ⇒x24+(y−1)24=1  ⇒(x2)2+(y−12)2=1  The parametric equation of circle is cos2t+sin2t=1  As the circle is moving once around it and clockwise , then 0≤t≤2π.  By comparing these two equations  x2=cos⁡t,y−12=−sin⁡  ⇒x=2cos⁡t,y−1=−2sin⁡  ⇒x=2cos⁡t,y=1−2sin⁡  Step 3  (b) The path of a particle that moves along the circle x2+(y−1)2=4 in the manner  three times around counterclockwise, starting at (2, 1).  x2+(y−1)2=4  The center of the circle is (0,1) and radius is 2   As the circle is moving thrice t is less than equal to thrice of 2π, then 0≤t≤6π  The parametric equation of circle is cos2t+sin2t=1.  By comparing the two equations   ⇒x2=cos⁡t,y−12=sin⁡  ⇒x=2cos⁡t,y−1=2sin⁡  ⇒x=2cos⁡t,y=1+2sin⁡.  Step 4   (c) The path of a particle that moves along the circle x2+(y−1)2=4 in the manner halfway around counterclockwise, starting at (0, 3).  x2+(y−1)2=4  The center of the circle is (0,1) and radius is 2   (0,3) lies on the y-axis.   As the circle is moving halfway around counterclockwise, starting at (0, 3), then  π2≤t≤π2+π  ⇒π2≤t≤3π2  The parametric equation of circle is cos2t+sin2t=1  By comparing the two equations  ⇒x2=cos⁡t,y−12=sin⁡  ⇒x=2cos⁡t,y−1=2sin⁡

Jeffrey Jordon · Accepted Answer

(a) The path of a particle which moves along the circle x2+(y−1)2=4 in such a manner that once around clockwise, starting at (2,1). Now converting the given equation x2+(y−1)2=4 into standard form as: x2+(y−1)2=4 x24+(y−1)24=1 (x2)2+(y−12)2=1 As it is known that the parametric equation of the circle is cos2⁡t+sin2⁡t=1. Now, it is given that circle is moving once around it and clockwise then: So, by comparing two equations cos2⁡t+sin2⁡t=1 and (x2)2+(y−12)2=1. the result received is: x2=cos⁡t⇒x=2cos y−12=−sin⁡t⇒y=1−2sin Therefore, the answer is x=2cos⁡t,y=1−2sin, where t∈[0,2π] (b) The path of a particle which moves along the circle x2+(y−1)2=4 in such a manner that two times around clockwise, starting at (2,1). The center of the circle is (0,1) and radius is 2. Now converting the given equation x2+(y−1)2=4 into standard form as: x2+(y−1)2=4 x24+(y−1)24=1 (x2)2+(y−12)2=1 As it is known that the parametric equation of the circle is cos2⁡t+sin2⁡t=1 As the circle is moving twice t is less than equal to twice of 2π, then: 0≤t≤4π So, by comparing two equations cos2⁡t+sin2⁡t=1 and (x2)2+(y−12)2=1, the result received is: x2=cos⁡t⇒x=2cos y−12=sin⁡t⇒y=1+2sin Therefore, the answer is x=2cos⁡t,y=1+2sin, where t∈[0,4π] (c) The path of a particle which moves along the circle x2+(y−1)2=4 in such a manner that half around counter-clockwise, starting at (0,3). The center of the circle is (0,1) and radius is 2 and (0,3) lies on y axis. Now converting the given equation x2+(y−1)2=4 into standard form as: x2+(y−1)2=4 x24+(y−1)24=1 (x2)2+(y−12)2=1 As it is known that the parametric equation of the circle is cos2⁡t+sin2⁡t=1 As the circle is moving halfway around counter-clockwise, starting at (0,3), then: 0≤t≤π So, by comparing two equations cos2⁡t+sin2⁡t=1 and (x2)2+(y−12)2=1, the result received is: x2=cos⁡t⇒x=2cos

Jazz Frenia · Accepted Answer

a) One clockwise rotation starting at (2,1):x(t)=2+2cos(t)y(t)=1+2sin(t) where 0≤t≤2π.b) Three counterclockwise rotations starting at (2,1):x(t)=2+2cos(3t)y(t)=1+2sin(3t) where 0≤t≤2π.c) Halfway counterclockwise rotation starting at (0,3):x(t)=2cos(t2)y(t)=3−2sin(t2) where 0≤t≤π.

Andre BalkonE · Accepted Answer

Step 1:a) To find the parametric equations for a particle moving clockwise along the circle x2+(y−1)2=4 starting at (2,1), we can use the angle θ as the parameter. Let&#039;s express x and y in terms of θ.The center of the circle is (0,1), and the radius is 2. Since the particle starts at (2,1), it is at an angle of π radians from the positive x-axis. We can use this information to find the parametric equations.First, we find x in terms of θ. The x-coordinate can be given as x=rcos(θ), where r is the radius. Substituting r=2, we have x=2cos(θ).Next, we find y in terms of θ. The y-coordinate can be given as y=rsin(θ), where r is the radius. Substituting r=2, we have y=2sin(θ).Therefore, the parametric equations for the particle moving clockwise along the circle are:x=2cos(θ)y=2sin(θ)where −π≤θ≤π.Step 2:b) To find the parametric equations for a particle moving counterclockwise along the circle x2+(y−1)2=4 starting at (2,1) and going three times around, we again use the angle θ as the parameter.Since the particle goes three times around counterclockwise, it will complete 3×2π=6π radians. We can express the angle θ as 0≤θ≤6π.The center of the circle is (0,1), and the radius is 2. Since the particle starts at (2,1), it is at an angle of π radians from the positive x-axis. We can use this information to find the parametric equations.Using the same approach as before, we find:x=2cos(θ)y=2sin(θ)where 0≤θ≤6π.Step 3:c) To find the parametric equations for a particle moving halfway around counterclockwise along the circle x2+(y−1)2=4 starting at (0,3), we again use the angle θ as the parameter.Since the particle moves halfway around counterclockwise, it will complete 12×2π=π radians. We can express the angle θ as 0≤θ≤π.The center of the circle is (0,1), and the radius is 2. Since the particle starts at (0,3), it is at an angle of π2 radians from the positive x-axis. We can use this information to find the parametric equations.Using the same approach as before, we find:x=2cos(θ)y=2sin(θ) where 0≤θ≤π.These are the parametric equations for the particle moving halfway around counterclockwise along the circle.

Find parametric equations for the path a particle that moves

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