Question

Find parametric equations for the path of a particle that moves along the circle x2 + (y − 3)2 = 16 in the manner described. (enter your answer as a comma-separated list of equations. let x and y be in terms of t.) (a) once around clockwise, starting at (4, 3). 0 ≤ t ≤ 2π.

Answer 1

Answer:

$x(t) = 4 cos(-t)\\y(t) = 4 sin(-t) + 3$

and t ∈ [0,2π]

Step-by-step explanation:

The standard equation for circle is:

$(x-h) ^2 + (y-k)^2 = r^2$

Comparing our equation with standard

$x^2 + (y-3)^2 =16$

h= 0,

k= 3

r= 4 (as in standard r², so r =√ r²)

Required:

Parametric equations of the circle

Formula:

$x = r cos(t) + h\\y= r cos(t) +k$

Putting values of r and h and k we get

$x= 4 cos(t) + 0 \\x= 4 cost(t) \\y= 4 sin(t) + 3$

As we need to start the object at (4,3)

x(0) = 4 and y(0) = 3

and the period is 2π

As the rotation is starting clock wise we will invert the value of t i.e -t

$x(t) = 4 cos(-t)\\y(t) = 4 sin(-t) + 3$

and t ∈ [0,2π]