Find the equation of the circle concentric With the circle `x^(2) + y^(2) - 4x - 6y - 9 = 0 ` and passing through the point ( -4, -5).

Answer

To find the equation of the circle that is concentric with the given circle and passes through the point (-4, -5), we can follow these steps:

### Step 1: Identify the center of the given circle
The equation of the given circle is:
\[ x^2 + y^2 - 4x - 6y - 9 = 0 \]

To find the center, we can rewrite this equation in the standard form by completing the square.

1. Rearranging the equation:
\[ x^2 - 4x + y^2 - 6y = 9 \]

2. Completing the square for \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

3. Completing the square for \(y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

4. Substituting back into the equation:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 = 9 \]
\[ (x - 2)^2 + (y - 3)^2 - 13 = 9 \]
\[ (x - 2)^2 + (y - 3)^2 = 22 \]

From this, we see that the center of the circle is at \( (2, 3) \).

### Step 2: Determine the radius of the required circle
Since the required circle is concentric with the given circle, it will have the same center \( (2, 3) \). We need to find the radius of the required circle, which passes through the point (-4, -5).

1. Calculate the distance from the center \( (2, 3) \) to the point \( (-4, -5) \):
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (-4, -5) \).

Substituting the values:
\[ r = \sqrt{((-4) - 2)^2 + ((-5) - 3)^2} \]
\[ = \sqrt{(-6)^2 + (-8)^2} \]
\[ = \sqrt{36 + 64} \]
\[ = \sqrt{100} = 10 \]

### Step 3: Write the equation of the required circle
Now that we have the center \( (2, 3) \) and the radius \( r = 10 \), we can write the equation of the circle in standard form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \( (h, k) = (2, 3) \) and \( r = 10 \).

Substituting the values:
\[ (x - 2)^2 + (y - 3)^2 = 10^2 \]
\[ (x - 2)^2 + (y - 3)^2 = 100 \]

### Final Equation
Thus, the equation of the required circle is:
\[ (x - 2)^2 + (y - 3)^2 = 100 \]

---