The circle `x ^(2) + y ^(2) =9` is contained in the circle `x ^(2) + y ^(2) - 6x - 8y + 25=c^(2)` If

Options

A. `c=2`
B. `c=3`
C. `c=5`
D. `c=10`

Answer

To determine the value of \( c \) such that the circle \( x^2 + y^2 = 9 \) is contained within the circle defined by the equation \( x^2 + y^2 - 6x - 8y + 25 = c^2 \), we will follow these steps:

### Step 1: Identify the first circle's properties
The first circle is given by the equation:
\[ x^2 + y^2 = 9 \]
This can be rewritten in standard form as:
\[ (x - 0)^2 + (y - 0)^2 = 3^2 \]
From this, we can identify:
- Center \( (0, 0) \)
- Radius \( r_1 = 3 \)

### Step 2: Rewrite the second circle's equation
The second circle is given by:
\[ x^2 + y^2 - 6x - 8y + 25 = c^2 \]
We can rearrange this into standard form by completing the square.

1. Group the \( x \) terms and \( y \) terms:
\[ (x^2 - 6x) + (y^2 - 8y) + 25 = c^2 \]

2. Complete the square:
- For \( x^2 - 6x \):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
- For \( y^2 - 8y \):
\[ y^2 - 8y = (y - 4)^2 - 16 \]

3. Substitute back into the equation:
\[ (x - 3)^2 - 9 + (y - 4)^2 - 16 + 25 = c^2 \]
Simplifying gives:
\[ (x - 3)^2 + (y - 4)^2 = c^2 - 9 + 16 - 25 \]
\[ (x - 3)^2 + (y - 4)^2 = c^2 - 18 \]

### Step 3: Identify the second circle's properties
From the above equation, we can identify:
- Center \( (3, 4) \)
- Radius \( r_2 = \sqrt{c^2 - 18} \)

### Step 4: Establish the condition for containment
For the first circle to be contained within the second circle, the distance between the centers plus the radius of the first circle must be less than or equal to the radius of the second circle:
\[ d + r_1 \leq r_2 \]
Where \( d \) is the distance between the centers:
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, we have:
\[ 5 + 3 \leq \sqrt{c^2 - 18} \]
This simplifies to:
\[ 8 \leq \sqrt{c^2 - 18} \]

### Step 5: Square both sides to eliminate the square root
\[ 64 \leq c^2 - 18 \]
Rearranging gives:
\[ c^2 \geq 82 \]

### Step 6: Solve for \( c \)
Taking the square root of both sides:
\[ c \geq \sqrt{82} \approx 9.055 \]
Since \( c \) must be a positive value, we conclude:
\[ c \geq 10 \quad (\text{since } c \text{ must be an integer}) \]

### Conclusion
The minimum integer value for \( c \) that satisfies the condition is:
\[ \boxed{10} \]