This problem asks us to find the sum of coefficients $\alpha+\beta+\gamma$ for a given hyperbola equation, using information about its foci. We will systematically use the properties of hyperbolas and algebraic manipulation to solve this.

1. Determine the Hyperbola's Center and Orientation

The first crucial step in analyzing any conic section, especially when given its foci, is to determine its center and the orientation of its transverse axis.

• Key Concept: The center of a hyperbola is the midpoint of the segment connecting its two foci.
• Why this step? The center $(h,k)$ is a fundamental parameter in the standard equation of a hyperbola. Its coordinates are directly related to the coefficients $\alpha$ and $\beta$ in the given general