At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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.
[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.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.