Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.

Question

(a) sketch  the  curve  represented  by  the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular  equation  whose  graph  represents  the  curve. Adjust  the  domain  of  the  resulting  rectangular  equation  if necessary.

x = 4 cos θ

y = 2 sin θ

Answer 1

Step 1 :

(a)

The parametric equations are and .

Eliminate the parameter :

Consider .

Consider .

Pythagorean identity :

Substitute  and  in above equation.

.

The graph of the equation represents an ellipse.

Compare the above equation with standard form of ellipse .

Center : .

.

Step 2 :

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 .

Note that the elliptic curve is traced out counter clock wise as varies from .

Step 3 :

(b)

The rectangular equation is .     [From (a)]

Observe the graph of the equation, the domain set is .

Solution :

(a)

The graph of the parametric equations and is :

(b)

The rectangular equation is .

The domain set is .