Find the equation of the circle whose radius is 3 and which touches internally the circle `x^2+y^2-4x-6y=-12=0` at the point `(-1,-1)dot`

Options

A. `5x^(2)+5y^(2)+8x -14y-16=0`
B. `5x^(2)+5y^(2)-8x-14y-32=0`
C. `5x^(2)+5y^(2)-8x+14y-4=0`
D. `5x^(2)+5y6(2)+8x+14y+12=0`

Explanation

To find the equation of the circle whose radius is 3 and which touches internally the circle given by the equation \(x^2 + y^2 - 4x - 6y = -12\) at the point \((-1, -1)\), we can follow these steps:

### Step 1: Rewrite the given circle equation
The given equation of the circle is:
\[ x^2 + y^2 - 4x - 6y + 12 = 0 \]
We can rewrite it in standard form by completing the square.

### Step 2: Complete the square
For the \(x\) terms:
\[ x^2 - 4x \quad \text{can be written as} \quad (x-2)^2 - 4 \]
For the \(y\) terms:
\[ y^2 - 6y \quad \text{can be written as} \quad (y-3)^2 - 9 \]
Now substituting back into the equation:
\[ (x-2)^2 - 4 + (y-3)^2 - 9 + 12 = 0 \]
This simplifies to:
\[ (x-2)^2 + (y-3)^2 - 1 = 0 \]
So, the equation of the larger circle is:
\[ (x-2)^2 + (y-3)^2 = 1 \]
This means the center of the larger circle is at \((2, 3)\) and its radius is \(5\).

### Step 3: Determine the center of the smaller circle
Let the center of the smaller circle be \((h, k)\). Since the smaller circle touches the larger circle internally at the point \((-1, -1)\), we can use the distance between the centers and the radii to find \((h, k)\).

### Step 4: Use the distance formula
The distance \(AC\) between the centers \(A(2, 3)\) and \(B(h, k)\) is given by:
\[ AC = \sqrt{(h - 2)^2 + (k - 3)^2} \]
Since the smaller circle has a radius of \(3\) and the larger circle has a radius of \(5\), the distance \(AC\) is:
\[ AC = 5 - 3 = 2 \]
Thus, we have:
\[ \sqrt{(h - 2)^2 + (k - 3)^2} = 2 \]
Squaring both sides gives:
\[ (h - 2)^2 + (k - 3)^2 = 4 \]

### Step 5: Use the point of tangency
The smaller circle touches the larger circle at the point \((-1, -1)\). This means that the point \((-1, -1)\) lies on the smaller circle. Therefore, we can write:
\[ (-1 - h)^2 + (-1 - k)^2 = 3^2 \]
This simplifies to:
\[ (-1 - h)^2 + (-1 - k)^2 = 9 \]

### Step 6: Solve the system of equations
Now we have two equations:
1. \((h - 2)^2 + (k - 3)^2 = 4\)
2. \((-1 - h)^2 + (-1 - k)^2 = 9\)

Expanding both equations:
1. \(h^2 - 4h + 4 + k^2 - 6k + 9 = 4\) simplifies to \(h^2 + k^2 - 4h - 6k + 9 = 0\)
2. \(h^2 + 2h + 1 + k^2 + 2k + 1 = 9\) simplifies to \(h^2 + k^2 + 2h + 2k - 7 = 0\)

### Step 7: Substitute and solve for \(h\) and \(k\)
Now we can solve these two equations simultaneously to find \(h\) and \(k\).

After solving, we find:
\[ h = \frac{4}{5}, \quad k = \frac{7}{5} \]

### Step 8: Write the equation of the smaller circle
The equation of the smaller circle can be written as:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h\), \(k\), and \(r = 3\):
\[ \left(x - \frac{4}{5}\right)^2 + \left(y - \frac{7}{5}\right)^2 = 3^2 \]

### Step 9: Simplify the equation
Expanding this gives:
\[ \left(x - \frac{4}{5}\right)^2 + \left(y - \frac{7}{5}\right)^2 = 9 \]
This can be further simplified to:
\[ 5x^2 + 5y^2 - 8x - 14y - 32 = 0 \]

### Final Answer
The equation of the required circle is:
\[ 5x^2 + 5y^2 - 8x - 14y - 32 = 0 \]