 Solution

The symmetric coefficient matrix of the quadratic form isThus $\boxed{q_a\text{ is nondegenerate if and only if }a\ne 1.}$ To obtain its signature, put $u=(y+z)/2$ and $v=(y-z)/2$. Completing the square yieldsThe transformation from $(x,y,z)$ to $(u+x,v,x)$ is invertible. The inertia of a bilinear form is consequentlyIf signature means the signed difference $n_{+}-n_{-}$, its value is $+1$ for $a>1$ and $-1$ for $a<1$. At $a=1$ the diagonal form has one zero direction, confirming degeneracy.