The two circles $x^2+y^2=ax$ and $x^2+y^2=c^2$ ($c>0$) touch each other if:

Circle 1: $(x-a/2{)}^2+y^2=(a/2{)}^2$. Center $C_1(a/2,0)$, radius $r_1=|a|/2$.Circle 2: $x^2+y^2=c^2$. Center $C_2(0,0)$, radius $r_2=c$.Distance between centers $C_1{C}_2=\sqrt{(a/2{)}^2+0}=|a|/2$.Circles touch if distance between centers equals $|r_1\pm r_2|$.$|a|/2=||a|/2\pm c|$ If $|a|/2=|a|/2+c$, then $c=0$ (impossible as $c>0$).If $|a|/2=|c-|a|/2|$, then:Either $|a|/2=c-|a|/2⟹|a|=c$ Or $|a|/2=|a|/2-c$ (impossible).So $|a|=c$.