The trajectory of a particle can be determined by analyzing its position coordinates with respect to time. In this case, the position coordinates of the particle are given by y(t) = A cos(2ωt) and x(t) = sin(ωt), where A is the amplitude of the motion and ω is the angular frequency.

To determine the trajectory of the particle, we can plot the position coordinates in a Cartesian coordinate system and observe the shape formed by the particle's motion.

1. Plotting the position coordinates:
Let's plot the position coordinates y(t) and x(t) on a graph.

On the y-axis, the position coordinate y(t) is given by A cos(2ωt).
On the x-axis, the position coordinate x(t) is given by sin(ωt).

2. Analyzing the shape:
By observing the plotted graph, we can determine the shape of the trajectory formed by the particle's motion.

a. Circle:
A circle is a closed curve in which all points are equidistant from a fixed center point. To determine if the trajectory is a circle, we need to check if the plotted graph forms a closed curve and if all points are equidistant from a fixed center point.

b. Ellipse:
An ellipse is a closed curve in which the sum of the distances from any point on the curve to two fixed points (called foci) is constant. To determine if the trajectory is an ellipse, we need to check if the plotted graph forms a closed curve and if the sum of the distances from any point on the curve to two fixed points is constant.

c. Hyperbola:
A hyperbola is a curve in which the difference of the distances from any point on the curve to two fixed points (called foci) is constant. To determine if the trajectory is a hyperbola, we need to check if the plotted graph forms a curve with two branches and if the difference of the distances from any point on the curve to two fixed points is constant.

d. Parabola:
A parabola is a curve in which all points are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). To determine if the trajectory is a parabola, we need to check if the plotted graph forms a curve with a single branch and if all points are equidistant from a fixed point and a fixed line.

3. Conclusion:
By analyzing the position coordinates and the shape formed by the plotted graph, we can conclude that the trajectory of the particle is a circle. The position coordinates y(t) = A cos(2ωt) and x(t) = sin(ωt) represent simple harmonic motion in two perpendicular directions. The resulting motion traces out a circle with a radius of A.