$\text{ Number of integral values of }b\text{ for which tangent parallel to line }y=x+1\text{ can be drawn to hyperbola }\frac{x^2}{5}-\frac{y^2}{b^2}=1 \text{is}$

$\text{ Equation of tangent to hyperbola }\frac{x^2}{5}-\frac{y^2}{b^2}=1\text{ having slope }m\text{ is }$

$\begin{array}{r}y=mx\pm \sqrt{a^2{m}^2-b^2}\\ \text{ or }y=x\pm \sqrt{5-b^2}\end{array}$

$\text{ Comparing it with }y=x+1\text{ , we get }$

$\begin{array}{r}{b}^2=1,4\\ \therefore b=\pm 1,\pm 2\end{array}$

$\text{ Thus, four integral values of }b\text{ are possible. }$