6. Given that the line connecting the vertex $V$ and the center $O$ of the base circle of a cone is perpendicular to the base, a plane passing through the midpoint $M$ of $VO$ is tangent to $\odot O$ and intersects the cone in an ellipse. If the radius of $\odot O$ is 1, then the length of the minor axis of the ellipse is ( ).
(A) $\frac{4 \sqrt{2}}{5}$
(B) $\frac{2 \sqrt{3}}{3}$
(C) $\frac{\sqrt{6}}{2}$
(D) None of the above
6.B.

Let the center of the ellipse be $S$, and the plane through the midpoint $M$ of $VO$ intersects $\odot O$ at point $A$. $MA$ intersects the ellipse at point $B$, then $AB$ is the major axis of the ellipse. Let $AC$ be the diameter of $\odot O$, then $O S \Perp \frac{1}{2} B C$. Therefore,
$$
S M=B M=\frac{1}{2} S A .
$$

Thus, $S$ is the centroid of $\triangle V O A$.
Connecting $VS$ intersects $AO$ at point $D$, and draw a chord $EF$ perpendicular to $OA$ through $D$, then the length of the minor axis of the ellipse is
$$
\frac{2}{3} E F=\frac{2}{3} \times 2 \sqrt{C D \cdot D A}=\frac{2 \sqrt{3}}{3} .
$$