Key Concepts and Formulas

• Standard Equation of a Circle: $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Minimum Distance Between Two Circles: If the distance between the centers is $d$, and the radii are $r_1$ and $r_2$, then the minimum distance between the circles is $d - r_1 - r_2$ when $d > r_1 + r_2$. If $d \le r_1 + r_2$, the circles intersect or touch, and the minimum distance is 0, unless one is completely contained inside the other.

Step-by-Step Solution

1. Convert Circle Equations to Standard Form to Find Centers and Radii

We need to rewrite the given equations in the standard form $(x-h)^2 + (y-k)^2 = r^2$ to determine the centers and radii of the circles. This involves completing the square for both $x$ and $y$ terms in each equation.

• Circle 1 ($S_1$): $x^2 + y^2 - 10x - 10y + 41 = 0$ Rearrange the terms: $(x^2 - 10x) + (y^2 - 10y) = -41$ Complete the square for the $x$ terms: $(x^2 - 10x + 25)$ requires adding 25. Complete the square for the $y$ terms: $(y^2 - 10y + 25)$ requires adding 25. Add 25 + 25 to both sides of the equation: $(x^2 - 10x + 25) + (y^2 - 10y + 25) = -41 + 25 + 25$ $(x - 5)^2 + (y - 5)^2 = 9$ Thus, the center of $S_1$ is $C_1 = (5, 5)$ and the radius is $r_1 = \sqrt{9} = 3$.

• Circle 2 ($S_2$): $x^2 + y^2 - 24x - 10y + 160 = 0$ Rearrange the terms: $(x^2 - 24x) + (y^2 - 10y) = -160$ Complete the square for the $x$ terms: $(x^2 - 24x + 144)$ requires adding 144. Complete the square for the $y$ terms: $(y^2 - 10y + 25)$ requires adding 25. Add 144 + 25 to both sides of the equation: $(x^2 - 24x + 144) + (y^2 - 10y + 25) = -160 + 144 + 25$ $(x - 12)^2 + (y - 5)^2 = 9$ Thus, the center of $S_2$ is $C_2 = (12, 5)$ and the radius is $r_2 = \sqrt{9} = 3$.

2. Calculate the Distance Between the Centers of the Circles

We use the distance formula to find the distance $d$ between the centers $C_1(5, 5)$ and $C_2(12, 5)$. $d = \sqrt{(12 - 5)^2 + (5 - 5)^2} = \sqrt{7^2 + 0^2} = \sqrt{49} = 7$ The distance between the centers is $d = 7$.

3. Determine the Relative Positions of the Circles

We compare the distance between the centers $d$ with the sum of the radii $r_1 + r_2$. $r_1 + r_2 = 3 + 3 = 6$. Since $d = 7 > 6 = r_1 + r_2$, the circles are external to each other; they do not intersect.

4. Calculate the Minimum Distance Between the Circles

Since the circles are external to each other, the minimum distance between them is given by: Minimum distance = $d - r_1 - r_2 = 7 - 3 - 3 = 1$.

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when completing the square and using the distance formula.
• Forgetting to Take the Square Root: Remember that the radius is the square root of the constant term on the right-hand side of the standard equation of a circle.
• Incorrectly Applying the Minimum Distance Formula: Always check the relative positions of the circles before applying $d - r_1 - r_2$. If the circles intersect or one is inside the other, this formula doesn't apply directly.

Summary

We first converted the given circle equations into standard form to find the centers and radii of the circles. Then, we calculated the distance between the centers and compared it with the sum of the radii to determine that the circles are external to each other. Finally, we used the formula $d - r_1 - r_2$ to find the minimum distance between the two circles, which is 1.

The final answer is $\boxed{1}$.