A technician is verifying a computer-aided design (CAD) file that uses the equation ${}5x^2 + 5y^2 - 20x + 10y - 15 = 0$ to define a circular part. To confirm that this equation represents a perfect circle, the technician must identify the specific requirement for the squared terms. According to the standard rule for circles, which characteristic of the coefficients for $x^2$ and $y^2$ confirms the shape is a circle?

A specialized design technician is reviewing technical specifications for a machine part. If the path of a cutting tool is defined by the equation ${}8x^2 + 8y^2 - 16x + 32y - 10 = 0$, this path represents a circle because the coefficients of the $x^2$ and $y^2$ terms are identical.

Verifying CAD Geometry for Circular Components

In a robotic assembly line, a circular tool path is modeled by a general quadratic equation. For the system to correctly identify the trajectory as a circle, the coefficients of the $x^2$ and $y^2$ terms must be ____.

A design engineer is auditing a list of geometric path equations for a robotic assembly system. Match each equation with the correct classification based on the standard rules for identifying a circle.