A design engineer is identifying geometric paths for a structural project. To confirm that a specific curve follows a hyperbolic path, the engineer reviews its general equation: ${}9x^2 - 16y^2 + 18x + 64y - 199 = 0$. Which feature of the squared terms in this equation identifies it as a hyperbola?

An architectural designer is evaluating a structural curve defined by the equation ${}25x^2 - 16y^2 + 50x + 64y - 439 = 0$. True or False: This equation represents a hyperbola because it contains both ${}x^2$ and ${}y^2$ terms that have different signs and different numerical coefficients.

Identification Criteria for Hyperbolic Curves

A design technician is using a CAD (Computer-Aided Design) software tool to model the steel supports for a bridge. To ensure the software correctly categorizes a curve as a hyperbola, the technician must verify the properties of the squared terms in the general equation. Match each characteristic of the equation with the requirement that identifies it as a hyperbola.

A software developer is writing an algorithm for a structural drafting program that categorizes conic sections based on their general equations. To ensure the program correctly identifies a hyperbola, the algorithm checks that the equation contains both $x^2$ and $y^2$ terms, and verifies that these squared terms have different numerical coefficients and different ____.