Step 1:

(a)

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

Eliminate the parameter $\"\"$ :

Consider $\"\"$ .

$\"\"$

Consider $\"\"$ .

$\"\"$

Pythagorean identity : $\"\"$

Substitute $\"\"$ and $\"\"$ in above equation.

$\"\"$

$\"\"$ .

Compare the above equation with standard form of ellipse $\"\"$ .

where $\"\"$ is the center of the ellipse, $\"\"$ is the length of the major axis and $\"\"$ is the length of the minor axis.

The distance between center and vertex is $\"\"$ .

The distance between center and focus is $\"\"$ .

$\"\"$ .

Center : $\"\"$

$\"\"$ .

$\"\"$

Thus, the graph of the equation represents an ellipse.

Graph the ellipse.

Graph the equation $\"\"$ .

Plot the center point $\"\"$ .

Plot the focus points $\"\"$ and $\"\"$ .

Plot the vertex points $\"\"$ and $\"\"$ .

Plot the two points above and below the center $\"\"$ and $\"\"$ .

$\"image\"$

Note that that elliptic curve is traced out counter clock wise as $\"\"$ varies from $\"\"$

The equation of an ellipse is $\"\"$ .

The curve equation is $\"\"$ .

Construct a table for different values of $\"\"$ .

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

$\"\"$	$\"\"$	$\"\"$	$\"\"$	$\"\"$	$\"\"$	$\"\"$	$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$	2	1.134	2	2.866	2	1.134	2

Graph:

(1) Graph the polar co-ordinates.

(2) Plot the points.

(3) Connect the points with a smooth curve.

Graph of the curve is $\"\"$ is :

$\"\"$

Step 1 :

(b)

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

Eliminate the parameter $\"\"$ :

Consider $\"\"$ .

$\"\"$

Consider $\"\"$ .

$\"\"$

Pythagorean identity : $\"\"$

Substitute $\"\"$ and $\"\"$ in above equation.

$\"\"$

Compare the above equation with standard form of ellipse $\"\"$ .

The graph of the equation represents an ellipse.

The equation of an ellipse is $\"\"$ .

Observe the graph of the equation, the domain set is $\"\"$ .