$\text{ If the line }y=x+2\text{ does not intersect any member of the family of the parabola }$$y^2=ax\left(a\in {\mathrm{ℝ}}^{+}\right)\text{at two distinct points then the maximum possible integral value of }$$\text{ latus-ractum of the parabola is }$

$\text{ Put }y=x+2 in y^2=ax\text{ we get }(x+2{)}^2-ax=0\Rightarrow x^2+(4-a)x+4=0$

$\text{ By given data discriminant of the above equation }\le 0$

$\Rightarrow (4-a{)}^2-4(4)\le 0$

$\Rightarrow a\le 8$

$\therefore \text{ Maximum possible integral value of latusrectum of the parabola is }8.$