In this write-up, I first use the parametric equations x =
acos(t) and y = bsin(t), where 0 < t < 6.28. I examine the
graphs of these equations when **a =
b**, **a < b**, and **a > b**. Then I move on to the
parametric equations x = acos(t) + hsin(t) and y = bsin(t) + hcos(t),
where 0 < t < 6.28, and I examine the graphs when **a = b**, **a
< b**, and **a > b**.

The following graph shows **a = b =
1**, **a = b = 2**, **a = b = 3**, **a
= b = 4**, and **a = b = 5**.

The following graph shows **a = b =
-1**, **a = b = -2**,
**a = b = -3**, **a
= b = -4**, and **a = b = -5**.

Notice that the graphs are all circles. Each graph crosses
the x-axis __and__ the y-axis at +/- a and +/-b. Putting a
negative on the a and b in the equation does not change the second
group of graphs from the first group; for example, the graph when
a = b = 1 is the same as the graph when a = b = -1.

Thus, I conclude that when | a | = | b |, the graph of the parametric equations in the form x = acos(t) and y = bsin(t) gives circles centered at the origin, with radii = | a | = | b |.

The following graph shows **a = 1 <
b = 2**, **a = 2 < b = 3**,
**a = 3 < b = 4**, **a = 4 < b = 5**, and **a
= 5 < b = 6** (Note in each case, a = b - 1).

The following graph shows **a = 1 <
b = 5**, **a = 2 < b = 6**,
**a = 3 < b = 7**, **a = 4 < b = 8**, and **a
= 5 < b = 9** (Note in each case, a = b - 4).

The following graph shows **a = -5 <
b = 1**, **a = -4 < b = 2**,
**a = -3 < b = 3**, **a = -2 < b = 4**, and **a
= -1 < b = 5** (Note in each case, a = b - 6).

Notice that the graphs are all ellipses.

Based on the first two examples, I would conjecture that each graph crosses the x-axis at +/- a and the y-axis at +/- b. Each graph has its center at the origin. So when a < b, it seems that the major axis lies on the y-axis, and the minor axis lies on the x-axis. But in the third example, there exist some negative values of a in the equations. In this case, the ellipses do not all have major axes on the y-axis and minor axes on the x-axis. So I must revisit my assumption.

It seems that the negative has no bearing on where the major and minor axes lie. In other words, when a = -5 and b = 1, although a < b, I know that | a | > | b |. Therefore, this graph does not have a major axis on the y-axis like the previous examples, but rather has a major axis on the x-axis instead. Also, when a = -3 and b = 3, although a < b, I know that | a | = | b |, and so this graph is a circle.

Thus, I need to make my conjecture more specific. I conclude that when | a | < | b |, the graph of the parametric equations in the form x = acos(t) and y = bsin(t) give ellipses centered at the origin, with a major axis on the y-axis and the minor axis on the x-axis.

The following graph shows **a = 2 >
b = 1**, **a = 3 > b = 2**,
**a = 4 > b = 3**, **a = 5 > b = 4**, and **a
= 6 > b = 5** (Note in each case, a = b + 1).

The following graph shows **a = 5 >
b = 1**, **a = 6 > b = 2**,
**a = 7 > b = 3**, **a = 8 > b = 4**, and **a
= 9 > b = 5** (Note in each case, a = b + 4).

The following graph shows **a = 1 >
b = -5**, **a = 2 > b = -4**,
**a = 3 > b = -3**, **a = 4 > b = -2**, and **a
= 5 > b = -1** (Note in each case, a = b + 6).

Notice that the graphs are all ellipses.

Based on the first two examples, I would conjecture that each graph crosses the x-axis at +/- a and the y-axis at +/- b. Each graph has its center at the origin. So when a > b, it seems that the major axis lies on the x-axis, and the minor axis lies on the y-axis. But in the third example, there exist some negative values of b in the equations. Now the ellipses do not all have major axes on the x-axis and minor axes on the y-axis. So I must revisit my assumption.

It seems that the negative has no bearing on where the major and minor axes lie. In other words, when a = 1 and b = -5, although a > b, I know that | a | < | b |. Therefore, this graph does not have a major axis on the x-axis like the previous examples, but rather has a major axis on the y-axis instead. Also, when a = 3 and b = -3, although a > b, I know that | a | = | b |, and so this graph is a circle.

Thus, I need to make my conjecture more specific. I conclude that when | a | > | b |, the graph of the parametric equations in the form x = acos(t) and y = bsin(t) give ellipses centered at the origin, with a major axis on the x-axis and the minor axis on the y-axis.

The following graph shows a = b = 2, and **h
= -2**, **h = -1**, **h = 0**, **h
= 1**, and **h = 2**.

Click **here** to see
a movie of the parametric equations x = 2cos(t) + hsin(t) and
y = 2sin(t) + hcos(t) as h varies from -10 to 10 and back.

I conjecture that the graphs can be a line, a circle, or an ellipse.

When | h | = a or | h | = b, the parametric equation graphs as a line.

When h = 0, the parametric equation graphs as a circle (in
this case, the equation reduces to x = a cos(t) and y = bsin(t),
with | a | = | b |, as **previously
discussed**).

In all other cases, the parametric equation seems to graph as an ellipse with the center at the origin. When h is negative, the ellipse has a major axis with a negative slope, and when h is positive, the ellipse has a major axis with a positive slope.

The following graph shows a = 2 < b = 5, and **h
= -2**, **h = -1**, **h = 0**, **h
= 1**, and **h = 2**.

Click **here** to see
a movie of the parametric equations x = 2cos(t) + hsin(t) and
y = 5sin(t) + hcos(t) as h varies from -10 to 10 and back.

I conjecture that the graphs are ellipses with the center at
the origin. When h is negative, the ellipse has a major axis with
a negative slope, and when h is positive, the ellipse has a major
axis with a positive slope. When h = 0, the ellipse has a major
axis that coincides with the y-axis (in this case, the equation
reduces to x = a cos(t) and y = bsin(t), with | a | < | b |,
as **previously discussed**).

The following graph shows a = 4 > b = 2, and **h
= -2**, **h = -1**, **h = 0**, **h
= 1**, and **h = 2**.

Click **here** to see
a movie of the parametric equations x = 4cos(t) + hsin(t) and
y = 2sin(t) + hcos(t) as h varies from -10 to 10 and back.

I conjecture that the graphs are ellipses with the center at
the origin. When h is negative, the ellipse has a major axis with
a negative slope, and when h is positive, the ellipse has a major
axis with a positive slope. When h = 0, the ellipse has a major
axis that coincides with the x-axis (in this case, the equation
reduces to x = a cos(t) and y = bsin(t), with | a | > | b |,
as **previously discussed**).