In Exercises 49–56, identify each equation without completing the square. 100x^2 - 7y^2 + 90y - 368 = 0

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The primary types include circles, ellipses, parabolas, and hyperbolas. Each type has a distinct standard form and properties, which help in identifying the nature of the equation. Understanding these forms is crucial for classifying the given equation.

Standard Form of Quadratic Equations

Quadratic equations can be expressed in standard form as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The coefficients A, B, C, D, E, and F determine the type of conic section represented by the equation. Recognizing the roles of these coefficients is essential for identifying the equation without completing the square.

Discriminant

The discriminant of a conic section is derived from the coefficients of the quadratic equation and is given by the formula D = B^2 - 4AC. This value helps determine the type of conic: if D < 0, it indicates an ellipse; if D = 0, a parabola; and if D > 0, a hyperbola. Analyzing the discriminant is a key step in classifying the given equation.