EMAT 6680

Assignment 2: Problem 10: Changing the Parameter of a
Conic Section

by

Thomas Cooper

Problem: Graph x^{2} + y^{2} = 9.

Now, on the same axes, graph x^{2} + xy
+ y^{2} = 9

Describe the new graph. Try different coefficients for the xy term. What kinds of graphs do you generate? What coefficients mark the boundaries between the different types of graphs? How do we know these are the boundaries? Describe what happens to the graph when the coefficient of the xy term is close to the boundaries.

This question is asking us to examine the effects of the parameter n on the
graph of x^{2} + nxy + y^{2}_{ }^{ }= 9.

Lets start with a few cases, including n = 0 and n = 1 as requested.

We see that x^{2} + y^{2} = 9 is a circle centered at (0,0) with radius 3.

Here we see that x^{2} + xy + y^{2}
= 9 is an ellipse, which passes through (3,0), (0,3),
(-3,0), and (0,-3). It is oriented along y=x and y=-x instead of the x and y
axes.

Lets try n = 2.

It appears that we now get 2 parallel lines. Why? Well x^{2} + 2xy + y^{2} = 9 is equivalent
to (x+y)^{2} = 9.

Taking the square root of both sides gives x + y = 3 or x + y= -3. So we indeed had y = -x + 3 and y = -x - 3.

What if we make n = 3?

Here we get a hyperbola with points (3,0), (0,3), (-3,0), and (0,-3) oriented along y = x and y = -x.

Note that it is easy to see by inspection that these four points will always
be on the graph of x^{2} + nxy + y^{2}
= 9.

With these cases in mind, lets look at an animation of x^{2} + nxy + y^{2} = 9 with graphing calculator 3.2.
(Click here)

So the graph is a circle when n = 0. As n increases, it stretches along y = -x as an ellipse until n = 2, where it becomes two parallel lines.

Beyond n = 2 it becomes a hyperbola being stretched in the direction of y = x.

If we let n be negative we will see that between 0 and -2 it is an ellipse being stretched along y = x. At n = -2, it is two parallel lines y = x + 3 and y = x - 3.

For an animation with n from -5 to 5, click here.

Can you prove these results with a rotation of the axes?