An ellipse passing through the origin has its foci (3, 4) and (6, 8). The length of its semi-minor axis is 6. Then the value of $\mathrm{b}/\sqrt{2}$is _____.

The points are A(3,4),B(6, 8), and O(0, 0).
OA + OB = 2a (where a is semi-major axis)
$$\begin{gathered} 2\mathrm{a}=5+10=15 \\ \therefore \mathrm{a}=\frac{15}{2} \\ \mathrm{Now}, 2\mathrm{ae}=\sqrt{(6-3{)}^2+(8-4{)}^2}=5 \\ \begin{array}{l}\therefore \mathrm{e}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{1}{3}\\ \therefore {\mathrm{b}}^2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{225}{4}\left(1-\frac{1}{9}\right)=50\end{array} \end{gathered}$$