If it is possible to draw the tangent to the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ having slope 2, then find its range of eccentricity.

For the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$

the tangent having slope m is  $\mathrm{y}=\mathrm{mx}\pm \sqrt{{\mathrm{a}}^2{\mathrm{m}}^2-{\mathrm{b}}^2}$

The tangent having slope 2 is $\mathrm{y}=2\mathrm{x}\pm \sqrt{4{\mathrm{a}}^2-{\mathrm{b}}^2}$

which is real if $4{\mathrm{a}}^2-{\mathrm{b}}^2\ge 0$ $\Rightarrow$ $\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}\le 4$

$\Rightarrow$ $$\begin{gathered} {\mathrm{e}}^2-1\le 4 \\ \text{ ⇒ }{\mathrm{e}}^2\le 5 \end{gathered}$$ 

$\Rightarrow$ $1<\mathrm{e}<\sqrt{5}$(for hyperbola e$>1)$