This equation represents a conic section, specifically a hyperbola. To determine the type of hyperbola, we can look at the coefficients of the x^2, xy, and y^2 terms.
The general equation of a hyperbola is of the form: Ax^2 + Bxy + Cy^2 = D
Comparing this general form with the given equation: A = 1, B = -4, C = -5, D = 1999
To determine the type of hyperbola, we need to calculate the discriminant (B^2 - 4AC): Discriminant = (-4)^2 - 4(1)(-5) = 16 + 20 = 36
Since the discriminant is positive, the hyperbola is real and has two branches. To determine whether it is a horizontal or vertical hyperbola, we can look at the signs of the coefficients:
If A and C have the same sign, the hyperbola is horizontal.If A and C have opposite signs, the hyperbola is vertical.
In this case, A = 1 (positive) and C = -5 (negative), so the hyperbola is vertical.
Therefore, the equation x^2 - 4xy - 5y^2 = 1999 represents a vertical hyperbola.
This equation represents a conic section, specifically a hyperbola. To determine the type of hyperbola, we can look at the coefficients of the x^2, xy, and y^2 terms.
The general equation of a hyperbola is of the form:
Ax^2 + Bxy + Cy^2 = D
Comparing this general form with the given equation:
A = 1, B = -4, C = -5, D = 1999
To determine the type of hyperbola, we need to calculate the discriminant (B^2 - 4AC):
Discriminant = (-4)^2 - 4(1)(-5) = 16 + 20 = 36
Since the discriminant is positive, the hyperbola is real and has two branches. To determine whether it is a horizontal or vertical hyperbola, we can look at the signs of the coefficients:
If A and C have the same sign, the hyperbola is horizontal.If A and C have opposite signs, the hyperbola is vertical.In this case, A = 1 (positive) and C = -5 (negative), so the hyperbola is vertical.
Therefore, the equation x^2 - 4xy - 5y^2 = 1999 represents a vertical hyperbola.