Let $A$ be a vertex of the ellipse $\mathrm{S}=\frac{{\mathrm{x}}^2}{4}+\frac{{\mathrm{y}}^2}{9}-1=0$ and  $F$ be a focus of the ellipse ${\mathrm{S}}^1=\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}-1=0.$ Let $P$ be a point on the major axis of the ellipse ${\mathrm{S}}^1=0,$which divides $\overline{\mathrm{OF}}$ in the ratio $2:1$ (O is the origin). If the length of the chord of the ellipse $S=0$ through $A$ and $P$ is $\frac{3\sqrt{101}}{\mathrm{k}}, \mathrm{then} \mathrm{k}=$