Find the equation of the circle which is concentric with `x^(2)+y^(2)+8x+12y+15=0` and passing through (2,3).

Answer

To find the equation of the circle that is concentric with the given circle \(x^2 + y^2 + 8x + 12y + 15 = 0\) and passes through the point (2, 3), we can follow these steps:

### Step 1: Identify the center of the given circle
The general form of the circle's equation is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the given equation, we can identify:
- \(2g = 8 \Rightarrow g = 4\)
- \(2f = 12 \Rightarrow f = 6\)

Thus, the center of the given circle is:
\[ (-g, -f) = (-4, -6) \]

### Step 2: Write the equation of the required circle
Since the required circle is concentric with the given circle, it will have the same center. Therefore, its equation will be of the form:
\[ x^2 + y^2 + 8x + 12y + c = 0 \]
where \(c\) is a constant that we need to determine.

### Step 3: Use the point (2, 3) to find \(c\)
Since the circle passes through the point (2, 3), we can substitute \(x = 2\) and \(y = 3\) into the equation:
\[ (2)^2 + (3)^2 + 8(2) + 12(3) + c = 0 \]
Calculating each term:
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(8(2) = 16\)
- \(12(3) = 36\)

Now, substituting these values into the equation:
\[ 4 + 9 + 16 + 36 + c = 0 \]
Simplifying:
\[ 65 + c = 0 \]
Thus, we find:
\[ c = -65 \]

### Step 4: Write the final equation of the required circle
Now substituting \(c\) back into the equation of the circle, we get:
\[ x^2 + y^2 + 8x + 12y - 65 = 0 \]

### Final Answer
The equation of the required circle is:
\[ x^2 + y^2 + 8x + 12y - 65 = 0 \]

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