Conic Explorations
By: Sydney Roberts
Exploring the graph of x2+xy+y2=9 for different coefficients of the xy term.
The equation for a circle
that is centered at the point (0,0) is x2+y2=r2
where r is the radius of the circle. Therefore, the equation x2+y2=9
gives us a circle centered at (0,0) with a radius of 3.
Now consider the graph x2+xy+y2=9.
What shape do you think this equation makes?
We could say that this
equation appears to give us an ellipse with the line y = -x being the major
axis.
But what if a coefficient
existed in front of the xy term? Then what would
happen? LetŐs explore with some different examples
Well it is pretty clear that
the general equation x2 + axy + y2
= 9 will not yield an ellipse for all values of a. So what values of a do give
us an ellipse? What other shapes will different values of a give us?
Well first, letŐs explore
what happens if a is negative.
Well we see that the negative
values give us the same shapes, except in a different direction. So maybe it is
only the absolute value of a that determines the shape
of these equations. Using the graph above, it appears that maybe when a = 2,
the shape of our graph is changing. Well letŐs explore this equation some. Note
that x2 + 2xy + y2 = 9 can be factored.
Which means that if a = 2,
then we get two linear functions (i.e. two lines).
However, if a 2, then what happens? Recall
that the standard formula for a conic is 
These conics are ellipses,
hyperbolas, or parabolas. To determine which shape the equation would yield, we
would need to look at the discriminant: B2 – 4AC.
á If the discriminant is less than 0, the function will
form an ellipse.
á If the discriminant is greater than 0, then the
function will form a hyperbola.
Well for all of our equations
that we are exploring,
A = C = 1, B varies, D=E=0,
and F = -9. Hence, the discriminant will be B2 – 4
for all of our equations (depending on B).
Hence, if |B| < 2, the
equation will form an ellipse, and if |B| > 2, then the equation will form a
hyperbola.