The circle concentric with `x^(2)+y^(2)+4x+6y+3=0` and radius 2 is

Options

A. `x^(2)+y^(2)+4x+6y-9=0`
B. `x^(2)+y^(2)+4x+6y+9=0`
C. `x^(2)+y^(2)-4x-6y+9=0`
D. `x^(2)+y^(2)=4`

Answer

To find the equation of the circle that is concentric with the given circle \( x^2 + y^2 + 4x + 6y + 3 = 0 \) and has a radius of 2, we will follow these steps:

### Step 1: Identify the center of the given circle
The equation of the circle is in the form:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \( g \) and \( f \) can be identified from the equation.

From the given equation:
- \( 2g = 4 \) → \( g = 2 \)
- \( 2f = 6 \) → \( f = 3 \)

The center of the circle is given by \( (-g, -f) \):
- Center = \( (-2, -3) \)

### Step 2: Write the equation of the new circle
Since the new circle is concentric with the given circle, it will have the same center \( (-2, -3) \) and a radius of 2.

The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

Substituting \( h = -2 \), \( k = -3 \), and \( r = 2 \):
\[ (x + 2)^2 + (y + 3)^2 = 2^2 \]
\[ (x + 2)^2 + (y + 3)^2 = 4 \]

### Step 3: Expand the equation
Now, we will expand the equation:
\[ (x + 2)^2 = x^2 + 4x + 4 \]
\[ (y + 3)^2 = y^2 + 6y + 9 \]

Combining these:
\[ x^2 + 4x + 4 + y^2 + 6y + 9 = 4 \]

### Step 4: Simplify the equation
Combine like terms:
\[ x^2 + y^2 + 4x + 6y + 13 = 4 \]

Now, rearranging gives:
\[ x^2 + y^2 + 4x + 6y + 13 - 4 = 0 \]
\[ x^2 + y^2 + 4x + 6y + 9 = 0 \]

### Conclusion
Thus, the equation of the circle that is concentric with the given circle and has a radius of 2 is:
\[ x^2 + y^2 + 4x + 6y + 9 = 0 \]