Where does a circle of radius R and center (A,A) intersect X axis. Where does a circle centered at (A,A) with radius R >= A intersect the X axis. (This is for a machining problem.)

Let $C$ be the circle, centered at $(a,a)$, with radius $r$, where $r \ge a >0$.

\begin{align*} \text{Then}\;&(x,0)\;\text{is on}\,C&&\\\\[6pt] \iff\; &(x-a)^2+(0-a)^2=r^2 &&\text{[by the distance formula]}\\\\[6pt] \iff\; &x^2 - 2ax + 2a^2 = r^2&&\\\\[6pt] \iff\; &x^2 - 2ax + a^2 = r^2-a^2&&\\\\[6pt] \iff\; &(x-a)^2 = r^2-a^2&&\\\\[6pt] \iff\; &x-a = \pm \sqrt{r^2-a^2}&&\\\\[6pt] \iff\; &x =a \pm \sqrt{r^2-a^2}&&\\\\[6pt] \end{align*}