The two circles x^2+y^2=ax and x^2+y^2=c^2 (c0) touch each other if:

Question

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

Accepted Answer

Circle 1: centre $\left(\tfrac a2,0\right)$, radius $\tfrac{|a|}{2}$. Circle 2: centre $(0,0)$, radius $c$.

Distance between centres $=\tfrac{|a|}{2}$. Since circle 1 passes through the origin (the centre of circle 2), the circles are internally tangent when
$$\frac{|a|}{2}=\left|c-\frac{|a|}{2}\right|\ \Rightarrow\ c=|a|\ \ (\text{taking the non-trivial branch}).$$

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