The two circles $x^2+y^2=ax$ and $x^2+y^2=c^2\text{ (with }c>0\text{ ) }$ touch each other, if .

The centres of the two circles are $C_1(-a/2,0)$  and $C_2(0,0)$, and their radii are $\frac{\left|a\right|}{2}$ and c. So, the two· circles 2 will touch each other, if 

$C_1{C}_2=$ Sum or difference of radii

$\begin{array}{l}\Rightarrow \sqrt{{\left(-\frac{a}{2}-0\right)}^2+(0-0{)}^2}=\left|c\pm \frac{\left|a\right|\mid }{2}\right|\\ \Rightarrow \frac{\left|a\right|}{2}=\mid c\pm \frac{\left|a\right|}{2\mid }\\ \Rightarrow c\pm \frac{\left|a\right|}{2}=\frac{\left|a\right|}{2}\end{array}$

$\Rightarrow c-\frac{\left|a\right|}{2}=\frac{\left|a\right|}{2}$ and $c+\frac{\left|a\right|}{2}=\frac{\left|a\right|}{2}$

$\Rightarrow c=\left|a\right|$ or, $c=0\Rightarrow c=\left|a\right|$