The two circles $x^2+y^2=ax$ and $x^2+y^2=c^2 \left(c>0\right)$ touch each other, if $\left|\frac{c}{a}\right|$ is equal to

The two circles touch each other internally as shown in the figure below

For the circle $x^2+y^2-2ax=0$, the centre is at $\left(\frac{a}{2},0\right)$ and radius is $\frac{a}{2}$ units.

Also, the circle is passing through the origin.

For the circle $x^2+y^2=a^2$, the centre is at $\left(0,0\right)$ and radius is $c$ units 

From the figure,

$c=2\left|\frac{a}{2}\right|=\left|a\right|$

Hence, $\left|\frac{c}{a}\right|=1$