How do you find parametric equations for the path of a particle that moves along the $x^2 + (y-3)^2 = 16$ once around clockwise, starting at (4,3) 0 ≤ t ≤ 2pi ?

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Answer 1 · 2016-08-07T15:49:47

$x = 4 cos t$
$y = 3 + 4 sin t$

$t in [0, 2 pi]$

this is a circle of radius 4 centred on (0,3)

we can use polar coordinates but with a twist to reflect the fact that the circle is not centred about the origin

thus we have

$x = 4 cos t$
$y = 3 + 4 sin t$

$implies sin^2 t + cos^2 t = (x/4)^2 + ((y-3)/4)^2 = 1$ !!

this satisfies the starting condition as $((x_o),(y_o)) = ((4 cos 0),(3 + 4 sin 0))= ((4),(3))$

it will move clockwise around the circle centre. in polar, angle t is measured clockwise from the x-axis.

How do you find parametric equations for the path of a particle that moves along the x^2 + (y-3)^2 = 16x^2 + (y-3)^2 = 16 once around clockwise, starting at (4,3) 0 ≤ t ≤ 2pi ?