Mathematics - Hyperbola Question with Solution | TestHub

${\mathrm{F}}_1,{\mathrm{F}}_2$ are left and right focus points of the hyperbola $\mathrm{C}:\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1(\mathrm{a}>0,\mathrm{b}>0)$. Point O is the origin. M is an arbitrary point on C and above the x -axis. H is a point on ${\mathrm{MF}}_1$. Given that ${\mathrm{MF}}_2\perp {\mathrm{F}}_1{\mathrm{F}}_2,{\mathrm{MF}}_1\perp \mathrm{OH},|\mathrm{OH}|=\lambda \left|{\mathrm{OF}}_2\right|$, where $\lambda \in \left(\frac{1}{3},\frac{1}{2}\right)$. Number of integers in the range of the eccentricity of the hyperbola C is $\_\_\_\_$