Find the equation of the circle concentric with the circle `x^(2) + y^(2) + 4x + 4y + 11 = 0 ` and passing through the point ( 5, 4).

Answer

To find the equation of the circle that is concentric with the given circle \( x^2 + y^2 + 4x + 4y + 11 = 0 \) and passes through the point \( (5, 4) \), we can follow these steps:

### Step 1: Identify the center of the given circle

The general form of a circle's equation is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the given equation \( x^2 + y^2 + 4x + 4y + 11 = 0 \), we can identify:
- \( 2g = 4 \) → \( g = 2 \)
- \( 2f = 4 \) → \( f = 2 \)
- \( c = 11 \)

The center of the circle is given by the coordinates \( (-g, -f) \):
\[ \text{Center} = (-2, -2) \]

### Step 2: Calculate the radius of the new circle

Since the new circle is concentric with the given circle, it shares the same center \( (-2, -2) \). To find the radius of the new circle, we need to calculate the distance from the center to the point \( (5, 4) \) using the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (-2, -2) \) and \( (x_2, y_2) = (5, 4) \):
\[ \text{Radius} = \sqrt{(5 - (-2))^2 + (4 - (-2))^2} \]
\[ = \sqrt{(5 + 2)^2 + (4 + 2)^2} \]
\[ = \sqrt{7^2 + 6^2} \]
\[ = \sqrt{49 + 36} \]
\[ = \sqrt{85} \]

### Step 3: Write the equation of the new circle

The equation of a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = -2 \), \( k = -2 \), and \( r = \sqrt{85} \):
\[ (x + 2)^2 + (y + 2)^2 = (\sqrt{85})^2 \]
\[ (x + 2)^2 + (y + 2)^2 = 85 \]

### Step 4: Expand the equation

Expanding the left side:
\[ (x^2 + 4x + 4) + (y^2 + 4y + 4) = 85 \]
Combining like terms:
\[ x^2 + y^2 + 4x + 4y + 8 = 85 \]
Subtracting 85 from both sides:
\[ x^2 + y^2 + 4x + 4y + 8 - 85 = 0 \]
\[ x^2 + y^2 + 4x + 4y - 77 = 0 \]

Thus, the equation of the required circle is:
\[ \boxed{x^2 + y^2 + 4x + 4y - 77 = 0} \]