A rectangle $ABCD$ has side lengths $AB=6 \text{ miles}$ and $BC=9\text{ miles}.$ A pigeon hovers at point $P$, which is 5 miles above some randomly chosen point inside $ABCD$. Given that the expected value of \[AP^2+CP^2-BP^2-DP^2\] can be expressed as $\tfrac{a}{b}$, what is $ab$? 

[i]2022 CCA Math Bonanza Lightning Round 2.2[/i]
1. **Identify the problem and given values:**
   - We have a rectangle \(ABCD\) with side lengths \(AB = 6\) miles and \(BC = 9\) miles.
   - A pigeon hovers at point \(P\), which is 5 miles above some randomly chosen point inside \(ABCD\).
   - We need to find the expected value of \(AP^2 + CP^2 - BP^2 - DP^2\).

2. **Apply the British Flag Theorem:**
   - The British Flag Theorem states that for any point \(P\) inside a rectangle \(ABCD\), the sum of the squares of the distances from \(P\) to two opposite corners is equal to the sum of the squares of the distances from \(P\) to the other two opposite corners.
   - Mathematically, this can be expressed as:
     \[
     AP^2 + CP^2 = BP^2 + DP^2
     \]
   - This implies that:
     \[
     AP^2 + CP^2 - BP^2 - DP^2 = 0
     \]

3. **Conclusion:**
   - Since the British Flag Theorem holds for any point \(P\) inside the rectangle, the expected value of \(AP^2 + CP^2 - BP^2 - DP^2\) is always \(0\).

The final answer is \(\boxed{0}\)