If two sets A and B are having 99 elements in common, the number of elements common to each of the sets `A xx B` and `B xx A` are `121 lambda^(2)`, the value of `lambda` is

Explanation

To solve the problem, we need to find the value of \( \lambda \) given that the number of elements common to the Cartesian products \( A \times B \) and \( B \times A \) is \( 121 \lambda^2 \) and that the two sets \( A \) and \( B \) have 99 elements in common.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The number of elements in the intersection of sets \( A \) and \( B \) is \( |A \cap B| = 99 \).
- The number of elements common to \( A \times B \) and \( B \times A \) is given as \( 121 \lambda^2 \).

2. **Understanding Cartesian Products**:
- The Cartesian product \( A \times B \) consists of ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \).
- The number of elements in \( A \times B \) is given by \( |A| \times |B| \).
- Similarly, the number of elements in \( B \times A \) is \( |B| \times |A| \).

3. **Common Elements in Cartesian Products**:
- The common elements in \( A \times B \) and \( B \times A \) are those pairs where the first element is from \( A \cap B \) and the second element is also from \( A \cap B \).
- Therefore, the number of common elements is given by \( |A \cap B| \times |A \cap B| = |A \cap B|^2 \).

4. **Setting Up the Equation**:
- From the above, we have:
\[ |A \cap B|^2 = 99^2 \]
- We also know that:
\[ |A \cap B|^2 = 121 \lambda^2 \]
- Setting these equal gives:
\[ 99^2 = 121 \lambda^2 \]

5. **Calculating \( 99^2 \)**:
- Calculate \( 99^2 \):
\[ 99^2 = 9801 \]

6. **Substituting into the Equation**:
- Substitute \( 9801 \) into the equation:
\[ 9801 = 121 \lambda^2 \]

7. **Solving for \( \lambda^2 \)**:
- Divide both sides by 121:
\[ \lambda^2 = \frac{9801}{121} \]

8. **Calculating \( \frac{9801}{121} \)**:
- Calculate \( \frac{9801}{121} \):
\[ \frac{9801}{121} = 81 \]

9. **Finding \( \lambda \)**:
- Taking the square root of both sides gives:
\[ \lambda = \sqrt{81} = 9 \]

### Final Answer:
Thus, the value of \( \lambda \) is \( 9 \).