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 distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• 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 must satisfy $|r_1 - r_2| < C_1C_2 < r_1 + r_2$.

Step-by-Step Solution

Step 1: Find the center and radius of the first circle.

The equation of the first circle is $x^2 + y^2 - 16x - 20y + 164 = r^2$. We need to rewrite this in standard form to identify its center and radius. We complete the square for both $x$ and $y$ terms.

$x^2 - 16x + y^2 - 20y = r^2 - 164$

To complete the square for $x^2 - 16x$, we add and subtract $(-16/2)^2 = 64$. To complete the square for $y^2 - 20y$, we add and subtract $(-20/2)^2 = 100$.

$x^2 - 16x + 64 + y^2 - 20y + 100 = r^2 - 164 + 64 + 100$ $(x - 8)^2 + (y - 10)^2 = r^2$

Thus, the center of the first circle is $C_1 = (8, 10)$ and its radius is $r_1 = r$.

Step 2: Find the center and radius of the second circle.

The equation of the second circle is $(x - 4)^2 + (y - 7)^2 = 36$. This is already in standard form.

Thus, the center of the second circle is $C_2 = (4, 7)$ and its radius is $r_2 = \sqrt{36} = 6$.

Step 3: Calculate the distance between the centers.

Using the distance formula, we find the distance between $C_1 = (8, 10)$ and $C_2 = (4, 7)$:

$C_1C_2 = \sqrt{(4 - 8)^2 + (7 - 10)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Step 4: Apply the intersection condition.

For the two circles to intersect at two distinct points, the following condition must hold:

$|r_1 - r_2| < C_1C_2 < r_1 + r_2$ Substituting the values we found: $|r - 6| < 5 < r + 6$

This gives us two inequalities:

1. $|r - 6| < 5$, which means $-5 < r - 6 < 5$. Adding 6 to all parts, we get $1 < r < 11$.
2. $5 < r + 6$, which means $r > -1$. Since the radius must be positive, this condition is always satisfied when $r > 1$.

Combining the inequalities, we have $1 < r < 11$.

Step 5: Verify the correct option.

The given correct answer is $r = 11$. Let's analyze why the solution is incorrect. We have $1 < r < 11$. However, $r=11$ would mean $|11-6| < 5 < 11+6 \implies 5<5<17$ which is not possible. Therefore, the given answer must be wrong. The condition for intersection is $|r_1 - r_2| < d < r_1 + r_2$. Substituting the values, $|r - 6| < 5 < r + 6$. This implies: $-5 < r - 6 < 5$ and $5 < r + 6$. $1 < r < 11$ and $r > -1$. Combining these two conditions, $1 < r < 11$. Therefore, the correct answer is $1 < r < 11$, which corresponds to option (D).

Common Mistakes & Tips

• Be careful when completing the square. Remember to add the same value to both sides of the equation.
• Remember the absolute value in the inequality $|r_1 - r_2| < C_1C_2 < r_1 + r_2$. This is crucial when you don't know which radius is larger.
• Always check if your final answer makes sense in the context of the problem. Radii must be positive.

Summary

We first found the centers and radii of the two circles by rewriting their equations in standard form. Then, we calculated the distance between the centers. Finally, we applied the condition for two circles to intersect at two distinct points, which led to the inequality $1 < r < 11$.

The final answer is \boxed{1 < r < 11}, which corresponds to option (D).