Which conic section is represented by the equation?
x^2 - y^2 + 24x + 4y + 28 = 0.
A hyperbola, because of the opposite signs of the x^2 and y^2 terms.
By completing squares, it can be rewritten in a standard form
[(x-a)/c]2 - [(y-b)/c]^2 = 1
To determine which conic section is represented by the equation: x^2 - y^2 + 24x + 4y + 28 = 0, we need to analyze its coefficients.
In general, the general equation of a conic section in standard form is given by:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Using the given equation, we can identify the coefficients:
A = 1
B = 0
C = -1
D = 24
E = 4
F = 28
Now, to determine the conic section, we can examine the sign and value of the coefficients.
For a conic section to be classified as a hyperbola, the coefficients A and C must have opposite signs, meaning one is positive and the other is negative.
In this case, A = 1 and C = -1, which satisfy this condition.
Furthermore, in a hyperbola, both the x^2 and y^2 terms have coefficients. However, in the given equation, the coefficient of the Bxy term is 0.
Hence, based on the absence of a Bxy term and the signs of A and C, we can conclude that the given equation represents a hyperbola.