Which equation describes a hyperbola?
A. 2x^2 + 2x + 2y^2 - 6y - 11 = 0
B. -6x^2 + 2x - 12y^2 - 11y = 11
C. -6y^2 + 2x - 11y = -5
D. 4y^2 + 2y - 10x^2 + 2x - 6 = 0
Thanks!
You should recognize certain patterns to the conic section equations.
if the signs of the x^2 and y^2 terms are opposite , then you have a hyperbola,
e.g. (your example)
If their signs are the same and their coefficients are also the same, you have a circle.
e.g. 6x^2 + 9x + 6y^2 - 4y = 123
If the signs are the same, but their coefficients are different, you have and ellipse
e.g. 6x^2 + 9x + 4y^2 - 4y = 123
if one of the square terms, either the x or the y, are missing, you have a parabola
4x^2 + 8y = 77
if both square terms are missing you have straight line.
To determine which equation describes a hyperbola, we need to look at the general form of the equation for a hyperbola.
The general form for a hyperbola centered at the origin is:
(x^2 / a^2) - (y^2 / b^2) = 1
where 'a' and 'b' are real numbers.
Let's analyze each given equation:
A. 2x^2 + 2x + 2y^2 - 6y - 11 = 0: This equation does not match the general form of a hyperbola.
B. -6x^2 + 2x - 12y^2 - 11y = 11: This equation does not match the general form of a hyperbola.
C. -6y^2 + 2x - 11y = -5: This equation does not match the general form of a hyperbola.
D. 4y^2 + 2y - 10x^2 + 2x - 6 = 0: This equation does not match the general form of a hyperbola.
None of the given equations matches the general form of a hyperbola, so the correct answer is none of the above.
To determine which equation describes a hyperbola, we need to analyze the equation and look for certain characteristics.
The standard form of a hyperbola equation is given by:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1 (for a horizontal hyperbola)
or
(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1 (for a vertical hyperbola)
Where (h, k) represents the center of the hyperbola and a and b are positive real numbers.
Let's analyze each given equation:
A. 2x^2 + 2x + 2y^2 - 6y - 11 = 0
This equation does not resemble the standard form of a hyperbola. The presence of terms with x^2 and y^2 coefficients equal to one indicates an ellipse rather than a hyperbola.
B. -6x^2 + 2x - 12y^2 - 11y = 11
Similar to option A, this equation does not match the standard form of a hyperbola. The presence of terms with x^2 and y^2 coefficients being negative indicates neither an ellipse nor a hyperbola.
C. -6y^2 + 2x - 11y = -5
Again, this equation does not resemble the standard form of a hyperbola. The absence of the necessary terms involving x makes it unlikely to be a hyperbola.
D. 4y^2 + 2y - 10x^2 + 2x - 6 = 0
This equation follows the pattern of a hyperbola. The presence of both x^2 and y^2 terms with coefficients of different signs is a characteristic of hyperbola equations.
Therefore, the equation that describes a hyperbola is D. 4y^2 + 2y - 10x^2 + 2x - 6 = 0.