= Solution
The symmetric coefficient <matrix> of the <quadratic form> is
$$
Q_a=\begin{pmatrix}a&1/2&1/2\\1/2&0&1/2\\1/2&1/2&0\end{pmatrix},\qquad \det Q_a=\frac{1-a}{4}.
$$
Thus $\boxed{q_a\text{ is nondegenerate if and only if }a\ne1.}$ To obtain its <signature>, put $u=(y+z)/2$ and $v=(y-z)/2$. Completing the square yields
$$
q_a=(u+x)^2-v^2+(a-1)x^2.
$$
The transformation from $(x,y,z)$ to $(u+x,v,x)$ is invertible. The <inertia of a bilinear form> is consequently
$$
\boxed{(n_+,n_-)=\begin{cases}(2,1)&a>1,\\(1,2)&a<1.\end{cases}}
$$
If 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.
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