(12) The minimum distance from the origin to a point on the curve $x^{2}-2 x y-3 y^{2}=1$ is $\qquad$ .
(12) $\frac{\sqrt{\sqrt{5}+1}}{2}$ Hint: Let the coordinates of a point on the curve be $M(r \cos \theta, r \sin \theta)$, then
$$
r^{2} \cos ^{2} \theta-2 r^{2} \sin \theta \cos \theta-3 r^{2} \sin ^{2} \theta=1,
$$

so
$$
\begin{aligned}
r^{2} & =\frac{1}{\cos ^{2} \theta-2 \sin \theta \cos \theta-3 \sin ^{2} \theta} \\
& =\frac{1}{-\sin 2 \theta+2 \cos 2 \theta-1} \\
& =\frac{1}{\sqrt{5} \sin (2 \theta+\varphi)-1} \\
& \geqslant \frac{1}{\sqrt{5}-1} \\
& =\frac{\sqrt{5}+1}{4} .
\end{aligned}
$$

Therefore, $r \geqslant \frac{\sqrt{\sqrt{5}+1}}{2}$.
(Supplied by Yan Hong, Xinjiang Experimental High School)