Write Up for Assignment #2

Problem #10

by

Jim Meneguzzo

In this problem we want to study the following graph:

Now let's look at another equation graphed on the same axis as the above equation:

As you can see the first equation produces a circle with center at the origin and radius of 3. The second equation produces a ellipse with it's x and y roots both having values of 3 and -3. Now let's examine what happens when we change the coefficient of the middle term of the ellipse as so:

The value on the right side of the equation has been changed to one to better illustrate what is happening to the graph, the only difference being now the circle has a radius of 1 and the ellipse has roots at 1 and -1. You can see that as we increase the value of the xy coefficient, the ellipse expands, yet maintains it's x and y roots of 1 and -1. If we investigate further, it can be shown that if the xy coefficient is allowed to continue to increase, there will be a point where the ellipse breaks into a hyperbola as we will now see:

The graph shows for xy coefficients >2, the ellipse will break into a hyperpola and the hyperbola will contract as the coefficients grow larger. It is interesting to note however, that the hyperbola's foci occur at 1,-1, the same as the circle and ellipse x,y roots. Now let's examine what happens when the xy coeffiecients are negative:

The negative coefficients do not affect the x and y roots as they are still 1 and -1. However, with the positive coefficients the upper side of the graph produces negative x and positive y values while the lower half produces the opposite positive x and negative y values as we would expect with the symmetry of the graphs. The negative coefficients produce both positive x and y values on the upper part of the graph and negative x and y values on the bottom part of the graph. This can be further illustrated with the following graph of both positive and negative xy coefficients: