CIRCLE AND ELLIPSE
By Dario Gonzalez Martinez
We will start considering the following parametric equation:
Our objective is to explore what curve (or curves) is associated to this parametric expression. We will explore how the variation of parameters a and b affects the shape of the curve related with the parametric expression. To this end, we will observe that the analysis is reduced to two board cases:
1) What is the shape of the curve when ?
2) What is the shape of the curve when ?
CASE 1:
For this case our initial expression is reduced to
We will attempt an initial graph for that expression when a = 1. This case is shows in figure 1 below:
Figure 1 
Figure 1 supports our perception that the curve should be a circle with center in the origin of the system and radius 1. This conjecture will become stronger if we observe the animation 1 below which shows how the shape of the curve change according to the variation of parameter a.

Animation 1 
The above animation suggests that when we vary parameter a, the shape of the curve is the same, a circle, the only difference is this circle has different radius, which seems to be equal to the value of a. Although we just made a conjecture, it is possible to prove that our conjecture is true. Let’s take a look at the figure 2 below:
Figure 2 
The circle with center O is a circle with radius a. We will denote with t the angle formed by segments a and x, and segments x and y are perpendicular to each other. We can use trigonometric relations to obtain
We also consider the trigonometric relation
By replacing the above relation in the
latter one, we have
This expression represents the equation of a circle with radius a and centered at the origin of the system. So, our conjecture is actually true.
CASE 2:
First, we will assume that a > b, so we start trying a = 2 and b =1 to graph an example that guide our analysis. In other words, we will consider the expression
In this manner, the curve will look like that in figure 3 below:
Figure 3 
Figure 3 above suggests several interesting facts. The curve associated to the expression seems to be an ellipse which has major axe equal to 2a = 2(2) = 4 and minor axes equal to 2b = 2(1) = 2. To make us sure about it, we should observe the following animations:


Animation 2(a) 
Animation 2(b) 
The above animations show how the curve behaves when parameters a and b vary. Animation 2(a) shows the variation of parameter a and b fixed, whereas animation 2(b) shows the variation of parameter b and a fixed.
Once again we can show that our conjecture is actually true. Consider the figure 4 below:
Figure 1 
The circles blue and green are concentric. A and B represent the intersections points of line OA and the blue and green circles, respectively. Lines AC and BD are perpendicular to xaxis, and the line defined by B and point (x , y) is parallel to xaxis. If we rotate line OA around center O, point (x , y) draws an ellipse with major axis the radius of the blue circle and minor axis the radius of green circle. We will consider as parameter t the angle COA.
From right triangles OAC and OBD we obtain
Therefore, the parametric equation for the locus of point (x , y) are
On the other hand, we know that
Which can be written as
follow
From which in turn we can obtain the canonic equation of an ellipse
Thus, our conjecture is true, and the
curve for the expression
Indeed represents an ellipse centered at
the origin system with major axe equal to 2a
and minor axe equal to 2b.
We could elaborate similar arguments to show
that the expression
Is also an ellipse centered at the origin
of the system when a < b.
The only difference is that this ellipse will be “taller” than “longer”. Figure 5 below shows an example for a = 1 and b = 2:
Figure 5 
FURTHER
ON THE ANALYSIS
Now we will consider the following
parametric equation
We will start considering a = b
=3, and we make h varies. We can observe
the behavior of the curve in the animation below:

Animation 3 
We can see that there is an instant, that
is, a value of h when the curve becomes into a line segment which passes through
the coordinate system origin, but the question is when this occurs. If we think about it we can conclude that a
line segment that passes through the coordinate system origin represents a
proportional relation between x and y. In
other words, we have to observe in which moment the system
becomes
linearly dependent. If we reorder the
system above, we will have
By using the determinant property of a
equation system
We can actually generalize this case for a = b = k, and we will obtain that
Thus, when h < k the curve becomes
more and more similar to the circle
The curve will be equal to the curve
above when h = 0. On the other hand, if
h > k, then the curve becomes more and more similar to the circle
The curve will be equal to the curve
above when h tends to infinite. This analysis
makes sense since we could consider the following:
On the other hand, we should also
consider the case when a
and b are different. The animation 4 below shows the case when a = 3 and b = 2, and we make h varies:

Animation 4 
The previous animation shows that the
curve is an ellipse centered at the coordinate system origin. Similarly to our previous analysis when a = b, the curve becomes into a linear
segment for some h, which divides our analysis “before” and “after” this
specific value of h. Likewise we did
before, we have to consider that a linear segment represents a proportional
relation between x and y, so the following equation system should be linearly
dependent
So, we should have that
or,
if we want to generalize this case
which means that h has to be equal to the geometric average between a and b. Therefore, we will have two cases “before” and “after” this specific value for h:

The curve becomes more and more similar to the ellipse: The curve is equal to the ellipse above when h = 0. 

The curve becomes more and more similar to the circle: The curve is equal to the circle above when h tends to the infinite 
Table 1 