Parametric Curves

Question

a. Describe the motion of a particle with position (x, y) as t varies in the given interval.
x = 4 + 3 cos t, y = 4 + 3 sin t, pi/2 ≤ t ≤ 3pi/2

 

b. Describe the motion of a particle with position (x, y) as t varies in the given interval.
x = 3 sin t, y = 5 + cos t, 0 ≤ t ≤ 3pi/2

 

Please explain, thank you in advance!

Answer 1

a. Note that (x-4)^2 + (y-4)^2 = 9, so the motion of the particle lies entirely on a circle of radius 3 centered at (4,4). As t ranges from pi/2 to 3pi/2, 3cos t ranges from 0 to -3 to 0, and 3sin t ranges from 3 to 0 to -3. So this is the left half of the circle, beginning at the top of the circle ((4,7)) and tracing its way at a consistent speed counter-clockwise down the left half, ending at the bottom of the circle ((4,1)).

 

b. Note that (x/3)² + [(y-5)/1]² = 1, so the motion of the particle lies entirely on an ellipse centered at (0,5) and having major axis from (-3,5) to (3,5) and minor axis from (0,6) to (0,4). As t ranges from 0 to 3pi/2, 3sin t ranges from 0 to 3 to 0 to -3, and cos t ranges from 1 to 0 to -1 to 0. So this is three-quarters of the ellipse, beginning at the top (0,6) and tracing clockwise around through the right-most point (3,5) then to the bottom (0,4) then to the left-most point (-3,5).

 

I hope that helps. In b I am using the standard equation of an ellipse ((x-x_0)/a)^2 + ((y-y_0)/b)^2 = 1, where the larger of a and b is the semi-major axis length and the smaller is the semi-minor axis length.