Coordinate Geometry: JEE Main 2025 Mathematics question

A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respectively. If the locus of the point $P$, that divides the rod $AB$ internally in the ratio $2:1$ is $9\left(x^2 + \alpha y^2 + \beta xy + \gamma x + 28 y\right) - 76 = 0$, then $\alpha - \beta - \gamma$ is equal to: