$\"\"$

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

Trigonometric identity : $\"\"$ .

Rewrite the equations

$\"\"$

Substitute $\"\"$ in the above equation

$\"\"$

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:

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

$\"\"$

Observe the graph:

From $\"\"$ to $\"\"$ , the parabola takes its first revolution in clockwise starting from $\"\"$ and again anti clockwise direction.

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

The particle moves clockwise and anti clockwise around the parabolic path.

$\"\"$

The particle moves counter clockwise and anti clockwise around the parabolic path.

The equation is $\"\"$ .