Write up #2

Exploring the equation of

by Kaori Tabeta

The Original Equations:Circle

Graph the equations of on the same axes is below.

It is clear that the graph of the equation of is circle. The graph of adding the term of -xy is ellipse.

The Observation of the graph of

Let's try different coeffcients for the xy term.

Now, the coeffcients for the xy term are -1,-2,-3,-4,-5 is below.

Then, there appear the ellipse and two lines and hyperbolas. We could say as following:

Temporarily the coeffcints of the xy term is n. As 0 > n > -2 , the graph is ellipse. As n=-2 , the graph is two lines. As n<-2 ,the graph is hyperbola.

More observe when 0>n>-2 as following:

Cange the ranges to look the boundaries as following:

Next, the coeffcients for the xy term are 1,2,3,4.

As 0<n<2, the graph is the ellipse. As n=2, the graph is the two lines. As n>2, the graph is hyperbora.

There are the four common points, (0,3),(0,-3),(3,0),(-3,0) ,on these graphs.

The Argebraic Argument

1. The four common points. Here

2. The graph of the equation of . Here