Now we will look at the cases when

a > b.

In this part of our investigation of parametric equations, we will be using the parametic equations

where 0 < t < 2Pi.

a + 1 = b

It is easy to conclude from the above sketch that each graph forms an ellipse that

crosses the x-axis at a positive/negative a value and a positive/negative b value and has its center at the origin.

We get the major axis lying on the x-axis, and the minor axis lying on the y-axis.

(Notice in yellow that when b = 0, you will get a horizontal line on the x-axis with a length of positive/negative a.)

a + 5 = b

Resulting in a positive b

We get the same sort of picture with these values, just a bit more exagerated horizontally,

because obviously the a value controls the width of the graph.

(Once again notice the b = 0 line)

a + 5 = b

Resulting in a negative b

But it is easy to see that this graph is very different from the first two.

The first thing that we notice is that we sometimes get the opposite results for the major and minor axes.

This forces us to take absolute value into consideration.

(Notice b = 0 and a = 0)

Conclusions:

When IaI > IbI , and a or b is not equal to zero, the graph of the parametric equation

results in ellipses centered at the origin, with a major axis on the x-axis and the minor axis on the y-axis.