did you guys get this : The x-component of motion of an object is given by x(t) = Axcos(ωxt + φx) and the y-component of motion of the object is given by y(t) = Aycos(ωyt + φy). What relationships between the A,ω, and φ parameters must be true so that the motion of the object is on a circle?

Aₓ = A_y = A

          wₓ = w_y = w

          Фₓ = Ф_y = Ф

Explanation:

For a movement to be circular it must meet the exception of the circle

          R² = x² + y²

in the exercise indicate the expressions of the movement in the two axes

        x (t) = Aₓ cos (wₓ t + Фₓ)

        y (t) = A_y cos (w_y t + Ф_y)

we substitute

         R² = Aₓ² cos²  (wₓ t + Фₓ) + A_y² sin²  w_y t + Ф_y)

for this expression to be a circle it must meet

          Aₓ = A_y = A

          wₓ = w_y = w

          Фₓ = Ф_y = Ф

with these expressions

           R² = A² [cos² (w t + Ф) + sin² (wₓ t + Фₓ) ]

use the trigonometry relationship

          cos² θ + sin² θ = 1

          R² = A²

Therefore, it is fulfilled that it is a circle whose radius is equal to the amplitude of the movement