Is 7x^2+7y^2-2xy=24 a hyperbola, a parabola, an ellipse, or a circle?

Question

If possible, tell how you were able to tell

Answer 1

This is the equation of an ellipse. If the xy term were not there, then you could say that this is a circle, as was suggested. However, as someone has already mentioned, the presence of the xy term means that there is a rotation of the axes (they didn't realize that this means this is not necessarily a circle).

There's a pretty good quick tutorial on this sort of thing at http://college.hmco.com/mathematics/larson/calculus_analytic/7e/shared/downloads/clc7eap0e01.pdf

The rotation here is given by, say, R, so that
cot(2R) = (7-7)/(-2) = 0
R = pi/4

Thus the substitutions we use are
x= sqrt(2)/2 * (x' - y')
y= sqrt(2)/2 * (x' + y')

Making these substitutions into the original equation, we have
6(x')^2 + 8(y')^2 = 24,
which is an equation of an ellipse.

Note: If you have your calculator (TI-89 or above) solve for y then graph the two solutions, you should be able to tell that this is not the equation of a circle. There is also a way to do this using linear algebra and matrices, but I've forgotten it, and this seems more straightforward.

Answer 2

The coefficients of the x² and y² terms are the same sign and equal. But that is not enought to conclude that it is a circle because there is also an xy term. To really be sure you need to take the discriminant.

Given the general form of a quadratic equation we have:

ax² + bxy + cy² + dx + ey + f = 0
7x² - 2xy + 7y² - 24 = 0

The discriminant is:

b² - 4ac = (-2)² - 4*7*7 = 4 - 196 = -192 < 0

Since the discriminant is less than zero it is the equation of an ellipse. And in addition, since a = c, the ellipse is a circle.

Answer 3

It's a circle. You can tell because both quadratic terms (x2 and y2) are positive and equal. The cross term -2xy rotates the axes by some amount that can be calculated, but that doesn't change the fact that the equation represents a circle.

Answer 4

circle because both verticies are seven an ellipse would have different coeficients for the x and y terms and a porabola only has one term that is squared but the 7x and 7y are squared

Answer 5

Circle

Answer 6

a circle.
given that it's not a hyperbola or a parabola, it's a circle because it has the factor 7 for both x and y.
it would be kind of you if i could get 10 points^_^