smc

$k$ will be between $1$ and $5$ for $AC+BC$ to be the smallest. If we mirror point $A$ across the y-axis to $A^{\prime }$, with coordinates $(-5,5),$ the distance $A^{\prime }C+BC$ will be same as $AC+BC$. The minimum of $A^{\prime }C+BC$ will occur when $C$ is on the straight line connecting $A^{\prime }$ and $B$ (i.e., $C$ lies on the line $A^{\prime }B$). Therefore, $k$ is the y-intercept of the line that passes through $A^{\prime }$ and $B$. The slope of the line is $\frac{1-5}{2-(-5)}=-\frac{4}{7}$. Using point-slope form, the equation of the line is $y-5=-\frac{4}{7}(x+5)$. Letting $x=0$ gives $y-5=-\frac{20}{7},$ so $y=\frac{15}{7}$. Therefore, $k=2\frac{1}{7}\Rightarrow \boxed{2\frac{1}{7}}.$