To determine which of the given equations represent hyperbolas, we should analyze the general form of a conic section equation:
[tex]\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \][/tex]
Hyperbolas have the property where the product of the coefficients of \( x^2 \) and \( y^2 \) is less than zero, i.e., \( A \cdot C < 0 \).
Let's examine each equation to identify the coefficients of \( x^2 \) (A) and \( y^2 \) (C):
1. \( 2x^2 + 16x + 2y^2 + 14y - 9 = 0 \)
- Coefficient of \( x^2 \) (A): 2
- Coefficient of \( y^2 \) (C): 2
- \( A \cdot C = 2 \cdot 2 = 4 \)
- Since \( 4 > 0 \), this is not a hyperbola.
2. \( 2x^2 + 4x - 5y^2 - 10y + 57 = 0 \)
- Coefficient of \( x^2 \) (A): 2
- Coefficient of \( y^2 \) (C): -5
- \( A \cdot C = 2 \cdot -5 = -10 \)
- Since \( -10 < 0 \), this equation represents a hyperbola.
3. \( -x^2 + 5x - 7y^2 + 2y - 81 = 0 \)
- Coefficient of \( x^2 \) (A): -1
- Coefficient of \( y^2 \) (C): -7
- \( A \cdot C = -1 \cdot -7 = 7 \)
- Since \( 7 > 0 \), this is not a hyperbola.
4. \( x - 2y^2 + 4y + 15 = 0 \)
- Coefficient of \( x^2 \) (A): 0 (which simplifies the equation)
- Since there is no \( x^2 \) term, this is not a hyperbola.
5. \( -x^2 + 12x + 3y^2 + 7y + 11 = 0 \)
- Coefficient of \( x^2 \) (A): -1
- Coefficient of \( y^2 \) (C): 3
- \( A \cdot C = -1 \cdot 3 = -3 \)
- Since \( -3 < 0 \), this equation represents a hyperbola.
Thus, the equations that represent hyperbolas are:
[tex]\[ 2x^2 + 4x - 5y^2 - 10y + 57 = 0 \][/tex]
[tex]\[ -x^2 + 12x + 3y^2 + 7y + 11 = 0 \][/tex]
Therefore, the correct answers are the second and fifth equations.
