The equation of a circle concentric with the circle `x^(2) + y^(2) - 6x + 12 y + 15 = 0 ` and double its area is

Options

A. A. `x^(2) + y^(2) - 6x + 12 y + 30 = 0 `
B. B. ` x^(2) + y^(2) - 6x + 12 y + 45 = 0`
C. C. ` x^(2) + y^(2) - 6x + 12 y - 30 = 0 `
D. D. `x^(2) + y^(2) - 6x + 12 y - 15 = 0 `

Answer

To solve the problem step by step, we need to find the equation of a circle that is concentric with the given circle and has double its area.

### Step 1: Identify the given circle's equation
The equation of the given circle is:
\[ x^2 + y^2 - 6x + 12y + 15 = 0 \]

### Step 2: Rewrite the equation in standard form
We can rearrange the equation to identify the center and radius. The general form of a circle's equation is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.

### Step 3: Find the center and radius
To find the center and radius, we can complete the square for the \(x\) and \(y\) terms.

1. For \(x\):
\[ x^2 - 6x \rightarrow (x - 3)^2 - 9 \]
2. For \(y\):
\[ y^2 + 12y \rightarrow (y + 6)^2 - 36 \]

Now substituting back into the equation:
\[ (x - 3)^2 - 9 + (y + 6)^2 - 36 + 15 = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y + 6)^2 - 30 = 0 \]
Thus, we can write it as:
\[ (x - 3)^2 + (y + 6)^2 = 30 \]
From this, we find:
- Center: \((3, -6)\)
- Radius: \(r = \sqrt{30}\)

### Step 4: Calculate the area of the given circle
The area \(A\) of a circle is given by:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (\sqrt{30})^2 = 30\pi \]

### Step 5: Determine the area of the new circle
Since the new circle has double the area of the given circle:
\[ \text{Area of new circle} = 2 \times 30\pi = 60\pi \]

### Step 6: Find the radius of the new circle
Let the radius of the new circle be \(R\):
\[ \pi R^2 = 60\pi \]
Dividing both sides by \(\pi\):
\[ R^2 = 60 \]
Thus, the radius \(R = \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}\).

### Step 7: Write the equation of the new circle
Using the center \((3, -6)\) and the radius \(R = \sqrt{60}\):
\[ (x - 3)^2 + (y + 6)^2 = 60 \]

### Step 8: Expand the equation
Expanding the equation:
\[ (x - 3)^2 + (y + 6)^2 = 60 \]
This gives:
\[ (x^2 - 6x + 9) + (y^2 + 12y + 36) = 60 \]
Combining like terms:
\[ x^2 + y^2 - 6x + 12y + 45 = 60 \]
Rearranging gives:
\[ x^2 + y^2 - 6x + 12y + 45 - 60 = 0 \]
Thus:
\[ x^2 + y^2 - 6x + 12y - 15 = 0 \]

### Final Answer
The equation of the new circle is:
\[ x^2 + y^2 - 6x + 12y - 15 = 0 \]