1) In the general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey = F = 0, what is the discriminant and what does it tell you about the graph of the equation?
2) Which terms in a general second-degree equation (Ax^2 + Bxy + Cy^2 + Dx + Ey = F = 0) are used to find the discriminant and determine what type of conic the equation represents?
1) The discriminant for a second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey = F = 0 can be calculated using the formula:
D = B^2 - 4AC
The discriminant tells us about the nature of the graph of the equation. Here's what different values of the discriminant indicate:
- If D > 0: The discriminant is positive, indicating that the equation represents a hyperbola. The graph will have two separate branches, and the curve will open either horizontally or vertically.
- If D = 0: The discriminant is zero, indicating that the equation represents a parabola. The graph will have a single curve, either opening upwards or downwards.
- If D < 0: The discriminant is negative, indicating that the equation represents an ellipse. The graph will be a closed curve, either elongated horizontally or vertically.
2) The discriminant formula mentioned above involves the coefficients of the second-degree terms (A, B, and C) in the equation. The discriminant itself helps determine the type of conic the equation represents based on its value, as discussed in the previous answer.