Key Concepts and Formulas

• The standard equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$.
• The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$.
• For two circles with centers $C_1, C_2$ and radii $r_1, r_2$ respectively, to intersect at two distinct points, the distance $d$ between their centers must satisfy $|r_1 - r_2| < d < r_1 + r_2$.

Step-by-Step Solution

1. Identify the Properties of the First Circle ($C_1$)

The equation of the first circle is given as $(x - 1)^2 + (y - 3)^2 = r^2$. We need to identify its center and radius by comparing it with the standard form of a circle's equation.

• Center of the first circle ($C_1$): $(1, 3)$
• Radius of the first circle ($r_1$): $r$

2. Identify the Properties of the Second Circle ($C_2$)

The equation of the second circle is given as $x^2 + y^2 - 8x + 2y + 8 = 0$. We need to rewrite this equation in standard form to find its center and radius. We can do this by completing the square.

$(x^2 - 8x) + (y^2 + 2y) + 8 = 0$ To complete the square for the $x$ terms, we need to add and subtract $(8/2)^2 = 16$. For the $y$ terms, we need to add and subtract $(2/2)^2 = 1$.

$(x^2 - 8x + 16) - 16 + (y^2 + 2y + 1) - 1 + 8 = 0$ $(x - 4)^2 + (y + 1)^2 - 16 - 1 + 8 = 0$ $(x - 4)^2 + (y + 1)^2 = 9$ $(x - 4)^2 + (y - (-1))^2 = 3^2$

• Center of the second circle ($C_2$): $(4, -1)$
• Radius of the second circle ($r_2$): $3$

3. Calculate the Distance Between the Centers ($d$)

We have the centers $C_1 = (1, 3)$ and $C_2 = (4, -1)$. We use the distance formula to find the distance $d$ between them.

$d = \sqrt{(4 - 1)^2 + (-1 - 3)^2}$ $d = \sqrt{(3)^2 + (-4)^2}$ $d = \sqrt{9 + 16}$ $d = \sqrt{25}$ $d = 5$

4. Apply the Condition for Two Distinct Intersection Points

The condition for two circles to intersect at two distinct points is $|r_1 - r_2| < d < r_1 + r_2$. We have $r_1 = r$, $r_2 = 3$, and $d = 5$. Substituting these values into the inequality, we get:

$|r - 3| < 5 < r + 3$

This inequality can be split into two separate inequalities:

(i) $5 < r + 3$ Subtracting 3 from both sides, we get: $2 < r$

(ii) $|r - 3| < 5$ This inequality means that $-5 < r - 3 < 5$. Adding 3 to all parts of the inequality, we get: $-2 < r < 8$

5. Combine the Results

We need to satisfy both conditions simultaneously: $2 < r$ and $-2 < r < 8$. Since $r$ represents the radius, $r$ must be positive. Combining the two inequalities, we have $2 < r < 8$.

6. Compare with Options

We found the condition $2 < r < 8$. Comparing this with the given options: (A) $r > 2$ (B) $2 < r < 8$ (C) $r < 2$ (D) $r = 2$

Option (A) $r>2$ is true, but it's not the tightest bound. Option (B) $2 < r < 8$ is the most precise condition that we derived.

Common Mistakes & Tips

• Remember that the radius of a circle must be positive. If you obtain a negative value or a range that includes negative values, double-check your calculations.
• Be careful when dealing with absolute value inequalities. Remember that $|x| < a$ is equivalent to $-a < x < a$.
• When completing the square, ensure that you add and subtract the correct value to maintain the equality.

Summary

We found the centers and radii of the two circles, calculated the distance between their centers, and applied the condition for two circles to intersect at two distinct points. This led us to the condition $2 < r < 8$. Therefore, the correct option is (B).

The final answer is \boxed{2 < r < 8}, which corresponds to option (B).