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To solve this problem, we need to find the equation of the circle given that the lines \( x + 2y + 3 = 0 \) and \( 2x + 3y + 7 = 0 \) are diameters of the circle and the area of the circle is \( \frac{198}{7} \) square units.

### Step 1: Find the center of the circle
The center of the circle lies at the intersection of the given lines, which can be found by solving the system of equations:

\[
\begin{aligned}
x + 2y + 3 &= 0 \quad \text{(Equation 1)}\\
2x + 3y + 7 &= 0 \quad \text{(Equation 2)}
\end{aligned}
\]

Multiplying Equation 1 by 2 to eliminate \( x \):

\[
2(x + 2y + 3) = 2x + 4y + 6 = 0 \quad \text{(Equation 3)}
\]

Subtracting Equation 2 from Equation 3:

\[
(2x + 4y + 6) - (2x + 3y + 7) = 0 \\
y - 1 = 0 \implies y = 1
\]

Substitute \( y = 1 \) into Equation 1:

\[
x + 2(1) + 3 = 0 \implies x + 5 = 0 \implies x = -5
\]

So, the center of the circle is \( (-5, 1) \).

### Step 2: Find the radius of the circle
The area of the circle is given by \( \frac{198}{7} \) square units. The area of a circle is also given by:

\[
\text{Area} = \pi r^2
\]

So,

\[
\pi r^2 = \frac{198}{7}
\]

\[
r^2 = \frac{198}{7\pi}
\]

### Step 3: Form the equation of the circle
The general equation of the circle is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Substituting \( h = -5 \) and \( k = 1 \):

\[
(x + 5)^2 + (y - 1)^2 = r^2
\]

Expanding and simplifying, we get:

\[
x^2 + 10x + 25 + y^2 - 2y + 1 = r^2 \\
x^2 + y^2 + 10x - 2y + 26 = r^2
\]

We know that \( r^2 = \frac{198}{7\pi} \). The equation should match one of the given options, where the constant term is represented.

Comparing the given options, the correct equation would match option **C**:

\[
x^2 + y^2 + 10x - 2y + 17 = 0
\]

### Conclusion
The correct answer is **C**.

Do you want any more details or have further questions?

Here are some related questions:
1. How do you determine the center of a circle given two diametric lines?
2. What is the geometric interpretation of a circle's equation?
3. How does the discriminant help identify the nature of a circle in an equation?
4. Can the area of a circle be used to find the radius in all cases?
5. What if the lines given were not diameters—how would you find the circle's equation?
6. How do you determine the radius from the equation of a circle?
7. What are the conditions for two lines to be diameters of a circle?
8. How does one verify the intersection point of two lines algebraically?

**Tip:** When solving for the intersection of two lines, substitution or elimination methods are very effective. Always double-check the solution by substituting back into the original equations.

Learn how to find the equation of a circle when given diametric lines x + 2y + 3 = 0 and 2x + 3y + 7 = 0. This detailed guide covers finding the center, calculating the radius using circle area, and forming the circle's equation. Suitable for Grades 10-12.

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