$\"\"$

The parametric equations are $\"\"$ and $\"\"$ and $\"\"$ .

Consider $\"\"$ .

$\"\"$

Similarly $\"\"$ .

Trigonometric identity : $\"\"$ .

Substitute $\"\"$ and $\"\"$ in the above identity.

$\"\"$

The above equation is in form of general form of ellipse.

So the particle moves in elliptical path.

$\"\"$

Draw a table for different values of $\"\"$ ranging from $\"\"$ .

Determine the direction of the curve.

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Graph:

(1) Draw the coordinate plane.

(2) Plot the points obtained in the table.

(3) Determine the directions of the curve.

$\"\"$

Observe the graph:

From $\"\"$ to $\"\"$ , the ellipse completes it first revolution in clockwise.

Similarly the ellipse completes it second and third revolution at $\"\"$ and $\"\"$ .

The particle moves in 3 times clockwise around the elliptical path.

$\"\"$

The particle moves in 3 times clockwise around the elliptical path.

The equation is $\"\"$ .