Question:

Step 1: Identify the center and radius of the first circle. The equation of the first circle is given as $(x - 1)^2 + (y - 3)^2 = r^2$. Comparing this with the standard form $(x - h)^2 + (y - k)^2 = R^2$, we can identify its center and radius. The center of the first circle is $C_1(1, 3)$ and its radius is $r_1 = r$. Step 2: Identify the center and radius of the second circle. The equation of the second circle is given as $x^2 + y^2 - 8x + 2y + 8 = 0$. Comparing this with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = -8 \Rightarrow g = -4$, $2f = 2 \Rightarrow f = 1$, and $c = 8$. The center of the second circle is $C_2(-g, -f) = (4, -1)$. The radius of the second circle is $r_2 = \sqrt{g^2 + f^2 - c}$. $$r_2 = \sqrt{(-4)^2 + (1)^2 - 8} = \sqrt{16 + 1 - 8} = \sqrt{9} = 3$$ So, the second circle has center $C_2(4, -1)$ and radius $r_2 = 3$. Step 3: State the condition for two circles to intersect at two distinct points. For two circles with centers $C_1$, $C_2$ and radii $r_1$, $r_2$ to intersect at two distinct points, the distance between their centers, $C_1C_2$, must satisfy the condition: $$|r_1 - r_2| < C_1C_2 < r_1 + r_2$$ Step 4: Calculate the distance between the centers of the two circles. Using the distance formula, $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, for $C_1(1, 3)$ and $C_2(4, -1)$: $$C_1C_2 = \sqrt{(4 - 1)^2 + (-1 - 3)^2} = \sqrt{(3)^2 + (-4)^2}$$ $$C_1C_2 = \sqrt{9 + 16} = \sqrt{25} = 5$$ The distance between the centers is $C_1C_2 = 5$. Step 5: Apply the condition and solve for the range of $r$. Substitute $r_1 = r$, $r_2 = 3$, and $C_1C_2 = 5$ into the condition from Step 3: $$|r - 3| < 5 < r + 3$$ This inequality can be split into two separate inequalities: 1) $|r - 3| < 5$ 2) $5 < r + 3$ Let's solve the first inequality: $|r - 3| < 5 \Rightarrow -5 < r - 3 < 5$ Adding 3 to all parts: $-5 + 3 < r < 5 + 3$ $-2 < r < 8$ Now, let's solve the second inequality: $5 < r + 3$ Subtracting 3 from both sides: $5 - 3 < r$ $2 < r$ Combining the results from both inequalities, $r$ must satisfy both $r > 2$ and $-2 < r < 8$. The intersection of these two ranges is $2 < r < 8$. Since the problem states $r > 0$, this condition is consistent with $2 < r < 8$. Step 6: Identify the odd positive integral values of $r$. We need to find positive integers $r$ such that $2 < r < 8$ and $r$ is odd. The integers in the range $(2, 8)$ are $3, 4, 5, 6, 7$. Among these integers, the odd positive values are $3, 5, 7$. Step 7: Count the number of such values. The odd positive integral values of $r$ are $3, 5, 7$. There are 3 such values. The final answer is $\boxed{3}$.