Line joining the points $(0,3)$ and $(5,-2)$ is a tangent to the curve $y=\frac{ax}{1+x}$, then

Equation of line joining the points $(0,3)$ and $(5,-2)$ is $y=3-x$.

If this line is tangent to $y=\frac{ax}{(x+1)}$, then

$(3-x)(x+1)=ax$ should have equal roots. 

Thus, $(a-2{)}^2+12=0$

$\Rightarrow$ No value of $a\Rightarrow a\in \varphi$