X^2+13y^2-6xy=100

This equation represents an ellipse. To determine the center, major and minor axes, and eccentricity of the ellipse, we can rewrite the equation in standard form.

First, let's rearrange the equation:
x^2 - 6xy + 13y^2 = 100

Next, we complete the square for both the x and y terms:
(x^2 - 6xy + 9y^2) + 4y^2 = 100 + 9y^2
(x - 3y)^2 + 4y^2 = 100 + 9y^2

Now, we can rewrite the equation as:
(x - 3y)^2 + 4y^2 = 100 + 9y^2

To determine the center, we set both terms equal to zero:
x - 3y = 0
x = 3y

So, the center of the ellipse is at (0, 0).

To determine the major and minor axes, we compare the coefficients of the x^2 and y^2 terms. Since the coefficient of x^2 is positive and greater than the coefficient of y^2, the major axis is along the x-axis.

The length of the major axis is given by 2a, where a is the square root of the value on the right side of the equation:
2a = sqrt(100 + 9y^2)
a = sqrt((100 + 9y^2)/4)

The length of the minor axis is given by 2b, where b is the square root of the value on the right side of the equation:
2b = sqrt(100 + 9y^2)
b = sqrt((100 + 9y^2)/4)

The eccentricity of the ellipse is given by the square root of (1 - (b^2/a^2)).

So, the center of the ellipse is (0, 0), the major axis has length 2a, the minor axis has length 2b, and the eccentricity is sqrt(1 - (b^2/a^2)).