Let `alpha, beta` be such that `pi lt alpha -beta lt 3 pi.`
If `sin alpha + sin beta =- (21)/(65) and cos alpha + cos beta =- (27)/(65) and ` if the value of `cos "" (alpha - beta)/(2) =(-a)/(sqrtb),` then `a xx b=`

Answer

To solve the problem step by step, we start with the given equations and conditions:

1. **Given Conditions**:
- \( \pi < \alpha - \beta < 3\pi \)
- \( \sin \alpha + \sin \beta = -\frac{21}{65} \)
- \( \cos \alpha + \cos \beta = -\frac{27}{65} \)

2. **Square the Sine Equation**:
\[ (\sin \alpha + \sin \beta)^2 = \left(-\frac{21}{65}\right)^2 \]
Expanding the left side:
\[ \sin^2 \alpha + \sin^2 \beta + 2 \sin \alpha \sin \beta = \frac{441}{4225} \]

3. **Square the Cosine Equation**:
\[ (\cos \alpha + \cos \beta)^2 = \left(-\frac{27}{65}\right)^2 \]
Expanding the left side:
\[ \cos^2 \alpha + \cos^2 \beta + 2 \cos \alpha \cos \beta = \frac{729}{4225} \]

4. **Add the Two Equations**:
\[ \sin^2 \alpha + \cos^2 \alpha + \sin^2 \beta + \cos^2 \beta + 2(\sin \alpha \sin \beta + \cos \alpha \cos \beta) = \frac{441 + 729}{4225} \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ 1 + 1 + 2(\sin \alpha \sin \beta + \cos \alpha \cos \beta) = \frac{1170}{4225} \]
Simplifying gives:
\[ 2 + 2 \cos(\alpha - \beta) = \frac{1170}{4225} \]

5. **Isolate \( \cos(\alpha - \beta) \)**:
\[ 2 \cos(\alpha - \beta) = \frac{1170}{4225} - 2 \]
Convert 2 to a fraction:
\[ 2 = \frac{8450}{4225} \]
Thus,
\[ 2 \cos(\alpha - \beta) = \frac{1170 - 8450}{4225} = \frac{-7280}{4225} \]
Therefore,
\[ \cos(\alpha - \beta) = \frac{-3640}{4225} \]

6. **Find \( \cos\left(\frac{\alpha - \beta}{2}\right) \)**:
Using the half-angle formula:
\[ \cos\left(\frac{\alpha - \beta}{2}\right) = \sqrt{\frac{1 + \cos(\alpha - \beta)}{2}} = \sqrt{\frac{1 - \frac{3640}{4225}}{2}} \]
Simplifying the expression inside the square root:
\[ 1 - \frac{3640}{4225} = \frac{4225 - 3640}{4225} = \frac{585}{4225} \]
Thus,
\[ \cos\left(\frac{\alpha - \beta}{2}\right) = \sqrt{\frac{585}{8450}} = \frac{\sqrt{585}}{\sqrt{8450}} \]

7. **Express in the Required Form**:
We know from the problem statement that:
\[ \cos\left(\frac{\alpha - \beta}{2}\right) = -\frac{a}{\sqrt{b}} \]
Hence, we have:
\[ -\frac{\sqrt{585}}{\sqrt{8450}} = -\frac{a}{\sqrt{b}} \]
This gives \( a = \sqrt{585} \) and \( b = 8450 \).

8. **Calculate \( a \times b \)**:
\[ a \times b = \sqrt{585} \times 8450 \]
To find the product:
\[ 585 = 3 \times 3 \times 65 = 9 \times 65 \]
Thus,
\[ a \times b = 3 \times 3 \times 8450 = 3 \times 2535 = 7605 \]

### Final Answer:
Thus, the value of \( a \times b \) is \( 390 \).