Investigate each of the following for .

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

, ,

**Consider**

If a=b, this parametric equation
show a circle with a radius=abs(a).
To explore Click here

**Why does the equation become
a circle?**
We know that and is
a equation of a circle with radius r.
Since
,
we can get a equation
i.e. .
If a>b, this parametric equation
show a ellipse like the follwing figure(a=6,
b=1)

To explore Click here

If a<b, this parametric equation
show a ellipse like the follwing figure(a=2,
b=4)

To explore Click here
**Why does these equations become
ellipses?**
We know that and is
a equation of an ellipse passing through (a,0), (-a,0), (0,b),
(-b,0).
Since
,
we can get a equation
i.e the equation of a ellaipse.

**Consider**

If a=b, this parametric equation
show a line y=-(x-a)+a in any one quadrant.
To explore Click here

If a>b or a<b, this parametric
equation show a line y=-(x-a)+b in one quadrant
determined by tha values of a and b.

The following is a graph when a=5, b=2.

To explore when a>=2 and b=2
Click here
This is a graph when a=2, b=5.

To explore in a case of a=2 and
b>=2 Click here
**Why does these equations become
lines?**
Since
and ,
we can get a line equation .
This eqation says that x-intercept is a and y-intercept
is b.
**Consider**

If a=b then
the graph is like the following(a=b=3 or a=b=-3)

To explore Click
here

If a>b then
the graph is like the following(a=5 or -5, b=2 or -2 )

To explore Click
here
If a<b then
the graph is like the following(a=2 or -2, b=5 or -5)

To explore Click
here
Let's graw the figures above simultaneously.

**My Conjecture**
is a curve shape **gradually
approaching to axises as n is increasing**.
If n is odd then the graph appears all quadrants and
if even then is does one quadrant dependinng on the values of
a and b.
We can investigate this cojecture by click
here.

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