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 given circles and determine the conditions under which they have common points. Here's the step-by-step solution:

### Step 1: Write the equations of the circles in standard form

The first circle is given by:
\[ x^2 + y^2 - 10x + 16y + 89 - r^2 = 0 \]

We can rearrange this to:
\[ x^2 - 10x + y^2 + 16y + (89 - r^2) = 0 \]

Now, we will complete the square for the \(x\) and \(y\) terms.

For \(x^2 - 10x\):
\[ x^2 - 10x = (x - 5)^2 - 25 \]

For \(y^2 + 16y\):
\[ y^2 + 16y = (y + 8)^2 - 64 \]

Substituting these back into the equation gives:
\[ (x - 5)^2 - 25 + (y + 8)^2 - 64 + (89 - r^2) = 0 \]
\[ (x - 5)^2 + (y + 8)^2 + (89 - 25 - 64 - r^2) = 0 \]
\[ (x - 5)^2 + (y + 8)^2 = r^2 \]

Thus, the center of the first circle is \((5, -8)\) and the radius is \(r\).

### Step 2: Write the second circle in standard form

The second circle is given by:
\[ x^2 + y^2 + 6x - 14y + 42 = 0 \]

Rearranging gives:
\[ x^2 + 6x + y^2 - 14y + 42 = 0 \]

Completing the square for \(x\) and \(y\):

For \(x^2 + 6x\):
\[ x^2 + 6x = (x + 3)^2 - 9 \]

For \(y^2 - 14y\):
\[ y^2 - 14y = (y - 7)^2 - 49 \]

Substituting these back gives:
\[ (x + 3)^2 - 9 + (y - 7)^2 - 49 + 42 = 0 \]
\[ (x + 3)^2 + (y - 7)^2 - 16 = 0 \]
\[ (x + 3)^2 + (y - 7)^2 = 16 \]

Thus, the center of the second circle is \((-3, 7)\) and the radius is \(4\).

### Step 3: Find the distance between the centers

The distance \(d\) between the centers \((5, -8)\) and \((-3, 7)\) is given by:
\[ d = \sqrt{(5 - (-3))^2 + (-8 - 7)^2} \]
\[ d = \sqrt{(5 + 3)^2 + (-8 - 7)^2} \]
\[ d = \sqrt{8^2 + (-15)^2} \]
\[ d = \sqrt{64 + 225} \]
\[ d = \sqrt{289} = 17 \]

### Step 4: Determine the conditions for common points

For the circles to have common points, the distance \(d\) must satisfy the following condition:
\[ |r - 4| \leq 17 \]

This can be split into two inequalities:
1. \(r - 4 \leq 17\)
2. \(- (r - 4) \leq 17\) or \(r - 4 \geq -17\)

Solving these inequalities:

1. From \(r - 4 \leq 17\):
\[ r \leq 21 \]

2. From \(r - 4 \geq -17\):
\[ r \geq -13 \]

Thus, we have:
\[ -13 \leq r \leq 21 \]

### Step 5: Count the integral values of \(r\)

The integral values of \(r\) in the range \([-13, 21]\) are:
\(-13, -12, -11, \ldots, 20, 21\).

To find the count:
- The smallest integer is \(-13\).
- The largest integer is \(21\).

The number of integers from \(-13\) to \(21\) is:
\[ 21 - (-13) + 1 = 21 + 13 + 1 = 35 \]

### Final Answer
The number of possible integral values of \(r\) is \(35\).

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