use the discriminant to determine whether the equation is a parabola, an ellipse, or a hyperbola. Then identify the angle of rotation required to eliminate the xy-term.

x^2+4xy+y^2=1

5x^2-6xy+5y^2-8x+8y-8=0

9x^2+24xy+16y^2=25

well, what are the discriminants? That is easy, and we can go from there.

To determine whether the given equations represent a parabola, an ellipse, or a hyperbola, we can use the discriminant. The discriminant is calculated based on the coefficients of the quadratic terms in each equation.

For the equation x^2 + 4xy + y^2 = 1:

1. Calculate the discriminant (D):

D = B^2 - 4AC

In this case, A = 1, B = 4, and C = 1:

D = (4^2) - 4(1)(1) = 16 - 4 = 12

2. Analyze the discriminant:

If the discriminant (D) is positive, it indicates an ellipse.
If the discriminant (D) is zero, it indicates a parabola.
If the discriminant (D) is negative, it indicates a hyperbola.

Since the discriminant (D) for the equation x^2 + 4xy + y^2 = 1 is positive (D = 12), it represents an ellipse.

To identify the angle of rotation required to eliminate the xy-term, we need to transform the given equation into a standard form by completing the square.

For the equation 5x^2 - 6xy + 5y^2 - 8x + 8y - 8 = 0:

1. Group the like terms:

(5x^2 - 6xy + 5y^2) - 8x + 8y - 8 = 0

2. Complete the square for the quadratic terms:

5(x^2 - (6/5)xy + (5/5)y^2) - 8x + 8y - 8 = 0
5(x^2 - (6/5)xy + (3/5)xy + (5/5)y^2) - 8x + 8y - 8 = 0
5(x - (3/5)y)^2 + 5/5y^2 - 8x + 8y - 8 = 0
5(x - (3/5)y)^2 + (1/5)y^2 - 8x + 8y - 8 = 0

3. Rearrange the terms:

5(x - (3/5)y)^2 + (1/5)y^2 - 8(x - 4y + 4) = 0
5(x - (3/5)y)^2 + (1/5)y^2 - 8(x - 4y) + 32 = 0

The equation can be written in the standard form as:

5(x - (3/5)y)^2 + (1/5)y^2 - 8(x - 4y) + 32 = 0

The angle of rotation required to eliminate the xy-term can be found by calculating the tangent of the angle using the coefficient of the xy-term.

For the equation 9x^2 + 24xy + 16y^2 = 25:

1. Calculate the coefficient of the xy-term:

The coefficient of the xy-term is 24.

2. Calculate the tangent of the angle:

Tangent of the angle = coefficient of xy-term / (coefficient of x^2 - coefficient of y^2)
Tangent of the angle = 24 / (9 - 16)
Tangent of the angle = -24 / 7

The angle of rotation required to eliminate the xy-term is the arctangent of -24/7.

In conclusion:
- The equation x^2 + 4xy + y^2 = 1 represents an ellipse.
- The equation 5x^2 - 6xy + 5y^2 - 8x + 8y - 8 = 0 requires further analysis to determine the angle of rotation.
- The equation 9x^2 + 24xy + 16y^2 = 25 requires further analysis to determine the angle of rotation.