Investigate each of the following for

Describe each when **a **= **b**, **a **< **b**,
and **a **> **b**.

Let's begin by examining

For values of

where **a **= **b**, **a **< **b**, and **a
**> **b**.

First, consider when **a = b** **= c** where **c**
is some constant.

Through examination of the parametric curve we can determine
that when **a=b=c** where **c** is a constant, then the
parametric curve will be a circle with x and y intercepts at **c**
and **-c**.

Next, let us examine the same parametric equations, but when
**a < b**, where **a=1** and **b=2, 3, 4, 5**. Then
the graph of the curve is

Now, through examination of the above parametric curve, we
can determine that when **a < b** that the curve will be
an ellipse whose major axis is on the vertical y-axis and minor
axis is on the horizontal x-axis. Additionally, the x-intercepts
will be **1** and **-1** (or **a** and **-a**) and
the y-intercepts will be **b** and **-b**.

Next, let us consider when **a > b** for the same parametric
equations with **b=1** and **a= 2, 3, 4, 5**. Then, the
graph of the curve is

Then, through examination of the parametric curve we then determine
that when **a > b** that the curve will be an ellipse whose
major axis is on the horizontal x-axis and minor axis on the vertical
y-axis. Additionally, the y-intercepts will be **1** and **-1**,
or **b** and **-b**, and the x-intercepts will be **a**
and **-a**.