Investigate the following equation when
for various values of a and b.
Let's take a look at the equation
when a = b, a < b, and a > b.



In this case, it is easy to see that when a=b, the outcome is a circle with its center at the origin. The radius of the circle is determined by the value of a and b. In other words, as a and b increase, the radius of the circle also increases. Actually, more specifically, a = b = the radius of the circle.
When a < b, what you get is an eclipse with its center at the origin. In this eclipse, a determines the width and b determines the height.
Similarly, when a > b, you also get an eclipse with its center at the origin. In this case, like the previous one, a also determines the width and b determines the height. So, this eclipse is rotated 90 degrees in relation to the previous case.
Let's look at some other cases of values of a and b, keeping one of the values constant while the other increases.


Again, we can see that when a = b (blue graph above), a circle is formed with radius equal to a and b.
Keeping a constant at 1 and increasing b, we see an eclipse that has a minor axis which remains constant along the xaxis, and a major axis that increases along the yaxis as b increases. The xintercepts remain 1 and 1, or more generally a and a. The yintercepts are always b and b.
The second graph above follows the same pattern, only this time b remains constant at 1. So, in this case, the minor axis is along the yaxis and remains constant. The major axis is along the xaxis and increases as a increases. The xintercepts are a and a, and the yintercepts are 1 and 1 (or b and b).