A design technician is verifying equations for a curved architectural feature. To ensure the equation represents an ellipse rather than another curved shape, what condition must the $x^2$ and $y^2$ terms satisfy in the general equation?

An architectural drafter is reviewing the mathematical specifications for a curved structural element. The element is defined by the equation ${}15x^2 + 15y^2 - 45x + 30y - 100 = 0$. The drafter concludes that this shape is an ellipse because both the $x^2$ and $y^2$ terms share the same positive sign. True or False: The drafter's identification is correct.

A quality control inspector at a precision lens manufacturing plant is reviewing the mathematical models for elliptical lens surfaces. Given a general equation such as ${}25x^2 + 16y^2 + 150x - 64y - 111 = 0$, match each component of the $x^2$ and $y^2$ terms to the specific requirement that identifies the shape as an ellipse.

Design Specification Verification

Architectural Geometry Audit

A CAD technician is reviewing a set of general equations used to model curved mechanical parts. To confirm that an equation represents an ellipse, the technician must verify that the $x^2$ and $y^2$ terms share the same sign but have ____ coefficients.

As a junior software developer at a structural engineering firm, you are writing a diagnostic script that classifies the shapes of load-bearing arches from their mathematical models. Arrange the logical steps your algorithm must execute in order to verify that a given general conic equation specifically represents an ellipse.