Find the equation of the circle concentric with the circle `x^(2) + y^(2) - 8x + 6y -5 = 0` and passing through the point (-2, -7).

Answer

To find the equation of the circle that is concentric with the given circle \( x^2 + y^2 - 8x + 6y - 5 = 0 \) and passes through the point (-2, -7), we can follow these steps:

### Step 1: Identify the center of the given circle
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the given equation \( x^2 + y^2 - 8x + 6y - 5 = 0 \), we can compare coefficients:
- \( 2g = -8 \) → \( g = -4 \)
- \( 2f = 6 \) → \( f = 3 \)

Thus, the center of the given circle is:
\[ (-g, -f) = (4, -3) \]

### Step 2: Write the general equation of the concentric circle
Since the required circle is concentric with the given circle, it will have the same center \( (4, -3) \). Therefore, the equation of the circle can be expressed as:
\[ (x - 4)^2 + (y + 3)^2 = r^2 \]
where \( r \) is the radius of the new circle.

### Step 3: Use the point (-2, -7) to find the radius
The new circle must pass through the point (-2, -7). We can substitute this point into the equation to find \( r^2 \):
\[ (-2 - 4)^2 + (-7 + 3)^2 = r^2 \]
Calculating the left-hand side:
\[ (-6)^2 + (-4)^2 = r^2 \]
\[ 36 + 16 = r^2 \]
\[ r^2 = 52 \]

### Step 4: Write the final equation of the circle
Now we can substitute \( r^2 \) back into the equation of the circle:
\[ (x - 4)^2 + (y + 3)^2 = 52 \]
Expanding this gives:
\[ (x^2 - 8x + 16) + (y^2 + 6y + 9) = 52 \]
Combining like terms:
\[ x^2 + y^2 - 8x + 6y + 25 = 52 \]
Subtracting 52 from both sides:
\[ x^2 + y^2 - 8x + 6y - 27 = 0 \]

### Conclusion
Thus, the equation of the circle that is concentric with the given circle and passes through the point (-2, -7) is:
\[ \boxed{x^2 + y^2 - 8x + 6y - 27 = 0} \]