Final answer:
To determine the conic section of the equation 21x² + 10√3xy + 31y² = 144, we compute the discriminant B² - 4AC using the given coefficients. The sign of the discriminant will indicate whether the graph represents an ellipse, a parabola, or a hyperbola.
Step-by-step explanation:
To determine whether the graph of the equation 21x² + 10√3xy + 31y² = 144 is a parabola, an ellipse, or a hyperbola, we use the discriminant of a conic section. For a general second-degree equation in the form Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant (D) can be found using the formula B² - 4AC. In this case, A = 21, B = 10√3, and C = 31.
The discriminant is then calculated: D = (10√3)² - 4(21)(31).
After calculating the discriminant, the graph will represent an ellipse if D is negative, a parabola if D is zero, and a hyperbola if D is positive. Therefore, by computing the value of D we can classify the conic section represented by the given equation.