If the circle `x^(2)+y^(2)-10x+16y+89-r^(2)=0` and `x^(2)+y^(2)+6x-14y+42=0` have common points, then the number of possible integral values of r is equal to

Options

A. 13
B. 14
C. 15
D. 18

Answer

To solve the problem, we need to analyze the two circles given by their equations and determine the conditions under which they have common points.

### Step-by-step Solution:

1. **Identify the first circle's equation:**
\[ x^2 + y^2 - 10x + 16y + 89 - r^2 = 0 \]
This can be rewritten in standard form by completing the square.

2. **Complete the square for the first circle:**
- For \(x\):
\[ x^2 - 10x = (x - 5)^2 - 25 \]
- For \(y\):
\[ y^2 + 16y = (y + 8)^2 - 64 \]
- Substitute back into the equation:
\[ (x - 5)^2 + (y + 8)^2 - 25 - 64 + 89 - r^2 = 0 \]
Simplifying gives:
\[ (x - 5)^2 + (y + 8)^2 = r^2 - 0 \]
- Thus, the center is \((5, -8)\) and the radius is \(|r|\).

3. **Identify the second circle's equation:**
\[ x^2 + y^2 + 6x - 14y + 42 = 0 \]
Again, we will complete the square.

4. **Complete the square for the second circle:**
- For \(x\):
\[ x^2 + 6x = (x + 3)^2 - 9 \]
- For \(y\):
\[ y^2 - 14y = (y - 7)^2 - 49 \]
- Substitute back into the equation:
\[ (x + 3)^2 - 9 + (y - 7)^2 - 49 + 42 = 0 \]
Simplifying gives:
\[ (x + 3)^2 + (y - 7)^2 = 16 \]
- Thus, the center is \((-3, 7)\) and the radius is \(4\).

5. **Calculate the distance between the centers:**
\[ \text{Distance} = \sqrt{(5 - (-3))^2 + (-8 - 7)^2} = \sqrt{(5 + 3)^2 + (-8 - 7)^2} = \sqrt{8^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \]

6. **Determine the conditions for the circles to intersect:**
The circles will intersect if:
\[ |r_1 - r_2| \leq d \leq r_1 + r_2 \]
Here, \(r_1 = |r|\) and \(r_2 = 4\), and \(d = 17\).

This gives us two inequalities:
- \( |r| - 4 \leq 17 \)
- \( |r| + 4 \geq 17 \)

7. **Solving the inequalities:**
- From \( |r| - 4 \leq 17 \):
\[ |r| \leq 21 \]
- From \( |r| + 4 \geq 17 \):
\[ |r| \geq 13 \]

8. **Combine the inequalities:**
\[ 13 \leq |r| \leq 21 \]

9. **Determine the integral values of \(r\):**
The possible integral values of \(r\) are:
\[ r = 13, 14, 15, 16, 17, 18, 19, 20, 21 \quad \text{and} \quad r = -13, -14, -15, -16, -17, -18, -19, -20, -21 \]
This gives us a total of \(9\) positive and \(9\) negative values, resulting in \(18\) integral values.

### Final Answer:
The number of possible integral values of \(r\) is \(18\).