26-15 Given that the origin is inside the ellipse $k^{2} x^{2}+y^{2}-4 k x+2 k y+k^{2}-1=0$, the range of the parameter $k$ is
(A) $|k|>1$.
(B) $|k| \neq 1$.
(C) $-1<k<1$.
(D) $0<|k|<1$.
(China High School Mathematics League, 1988)
[Solution] Let $f(x, y)=k^{2} x^{2}+y^{2}-4 k x+2 k y+k^{2}-1$, then when point $P\left(x_{0}, y_{0}\right)$ is inside the ellipse, we have $f\left(x_{0}, y_{0}\right)<0$.
$\therefore f(0,0)=k^{2}-1<0$, which means $0<|k|<1$.
Therefore, the answer is $(D)$.