1. Given that the origin is inside the circle $k^{2} x^{2}+y^{2}-4 k x+2 k y+k^{2}$ $-1=0$. Then, the range of values for the number $k$ is ( ).
(A) $|k|>1$
(B) $|k| \neq 1$
(C) $-1<k<1$
(D) $0<|k|<1$
To find the "criterion", we can use the "specialization" method: Given that the origin $(0,0)$ is inside the ellipse $f(x, y)=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-1=0$, we have $f(0,0)=-1 < 0$ (and vice versa); thus, we know that the origin being inside the ellipse $f(x, y)=0$ means $f(0,0)<0$. Let $f_{k}(x, y)$ be the original equation, then
$$
\left\{\begin{array}{l}
f(0,0)=k^{2}-1<0, \\
k \neq 0 .
\end{array}\right.
$$

It is clear that (D) is correct.