To solve the problem, we need to find the values of \(a\) and \(b\) such that the area of triangle \(AOB\) is 11, and then calculate \(4b^2 + 9a^2\).
### Step-by-Step Solution:
1. **Understanding the Area of Triangle AOB:**
The area of triangle \(AOB\) formed by points \(A(a, 0)\), \(B(0, b)\), and the origin \(O(0, 0)\) can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is \(a\) and the height is \(b\). Therefore, the area can be expressed as:
\[
\text{Area} = \frac{1}{2} \times a \times b
\]
2. **Setting Up the Equation:**
Given that the area is 11, we can set up the equation:
\[
\frac{1}{2} \times a \times b = 11
\]
Multiplying both sides by 2 gives:
\[
ab = 22 \quad \text{(Equation 1)}
\]
3. **Finding the Equation of the Line AB:**
The line \(AB\) passes through the point \(P(2, 3)\) and intersects the x-axis at \(A(a, 0)\) and the y-axis at \(B(0, b)\). The slope of line \(AB\) can be calculated as:
\[
\text{slope} = \frac{b - 0}{0 - a} = -\frac{b}{a}
\]
The slope of line \(PA\) is:
\[
\text{slope} = \frac{3 - 0}{2 - a} = \frac{3}{2 - a}
\]
4. **Setting the Slopes Equal:**
Since points \(A\), \(B\), and \(P\) are collinear, we can set the slopes equal:
\[
-\frac{b}{a} = \frac{3}{2 - a}
\]
Cross-multiplying gives:
\[
-b(2 - a) = 3a
\]
Expanding this, we have:
\[
-2b + ab = 3a \quad \text{(Equation 2)}
\]
5. **Rearranging Equation 2:**
Rearranging Equation 2 gives:
\[
ab - 3a + 2b = 0
\]
We can substitute \(ab = 22\) from Equation 1 into this equation:
\[
22 - 3a + 2b = 0
\]
Rearranging gives:
\[
2b = 3a - 22 \quad \text{(Equation 3)}
\]
6. **Substituting Equation 3 into Equation 1:**
From Equation 3, we can express \(b\) in terms of \(a\):
\[
b = \frac{3a - 22}{2}
\]
Substituting this into Equation 1:
\[
a \left(\frac{3a - 22}{2}\right) = 22
\]
Multiplying through by 2:
\[
a(3a - 22) = 44
\]
Expanding gives:
\[
3a^2 - 22a - 44 = 0
\]
7. **Solving the Quadratic Equation:**
We can use the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(A = 3\), \(B = -22\), and \(C = -44\):
\[
a = \frac{22 \pm \sqrt{(-22)^2 - 4 \cdot 3 \cdot (-44)}}{2 \cdot 3}
\]
Calculating the discriminant:
\[
= \frac{22 \pm \sqrt{484 + 528}}{6} = \frac{22 \pm \sqrt{1012}}{6}
\]
Simplifying gives:
\[
= \frac{22 \pm 2\sqrt{253}}{6} = \frac{11 \pm \sqrt{253}}{3}
\]
8. **Finding b:**
Substitute \(a\) back into Equation 3 to find \(b\):
\[
b = \frac{3\left(\frac{11 \pm \sqrt{253}}{3}\right) - 22}{2}
\]
Simplifying gives:
\[
b = \frac{11 \pm \sqrt{253} - 22}{2} = \frac{-11 \pm \sqrt{253}}{2}
\]
9. **Calculating \(4b^2 + 9a^2\):**
Now we can substitute \(a\) and \(b\) into \(4b^2 + 9a^2\):
\[
4b^2 = 4\left(\frac{-11 \pm \sqrt{253}}{2}\right)^2 = \frac{4(121 - 22\sqrt{253} + 253)}{4} = 374 - 22\sqrt{253}
\]
\[
9a^2 = 9\left(\frac{11 \pm \sqrt{253}}{3}\right)^2 = \frac{9(121 + 22\sqrt{253} + 253)}{9} = 374 + 22\sqrt{253}
\]
Adding them gives:
\[
4b^2 + 9a^2 = (374 - 22\sqrt{253}) + (374 + 22\sqrt{253}) = 748
\]
### Final Answer:
Thus, the value of \(4b^2 + 9a^2\) is:
\[
\boxed{748}
\]
