1. **Problem Statement:** Find the equation of a circle that touches the coordinate axes at points $(a,0)$ and $(0,a)$. 2. **Understanding the problem:** A circle touching the coordinate axes at $(a,0)$ and $(0,a)$ means these points lie on the circle and the circle is tangent to both axes. 3. **Formula and properties:** The general equation of a circle is $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h,k)$ is the center and $r$ is the radius. 4. Since the circle touches the x-axis at $(a,0)$, the distance from the center to the x-axis equals the radius. Similarly, it touches the y-axis at $(0,a)$, so the distance from the center to the y-axis equals the radius. 5. Let the center be $(h,k)$. Because the circle touches both axes, the radius $r$ equals both $|h|$ and $|k|$. Since the points are positive, $h = k = r$. 6. The circle passes through $(a,0)$, so substitute into the circle equation: $$ (a - r)^2 + (0 - r)^2 = r^2 $$ 7. Simplify: $$ (a - r)^2 + r^2 = r^2 $$ $$ (a - r)^2 = 0 $$ 8. This implies: $$ a - r = 0 \Rightarrow r = a $$ 9. Therefore, the center is at $(a,a)$ and radius $r = a$. 10. The equation of the circle is: $$ (x - a)^2 + (y - a)^2 = a^2 $$ **Final answer:** $$ (x - a)^2 + (y - a)^2 = a^2 $$