Let a,bgt0 and `alpha=(hati)/(a)+(4hatj)/(b)+bhatk` and `beta=bhati+ahattj+(1)/(b)hatk`, then the maximum value of `(10)/(5+alpha*beta)` is

Options

A. 1
B. 2
C. 4
D. 8

Answer

To solve the problem, we need to find the maximum value of the expression \( \frac{10}{5 + \alpha \cdot \beta} \), where \( \alpha \) and \( \beta \) are given vectors. Let's go through the steps to find this maximum value.

### Step 1: Define the vectors \( \alpha \) and \( \beta \)

Given:
\[ \alpha = \frac{\hat{i}}{a} + \frac{4\hat{j}}{b} + b\hat{k} \]
\[ \beta = b\hat{i} + a\hat{j} + \frac{1}{b}\hat{k} \]

### Step 2: Calculate the dot product \( \alpha \cdot \beta \)

The dot product of two vectors is calculated as follows:
\[ \alpha \cdot \beta = \left(\frac{1}{a}\hat{i} + \frac{4}{b}\hat{j} + b\hat{k}\right) \cdot \left(b\hat{i} + a\hat{j} + \frac{1}{b}\hat{k}\right) \]

Calculating the dot product:
\[ \alpha \cdot \beta = \left(\frac{1}{a} \cdot b\right) + \left(\frac{4}{b} \cdot a\right) + \left(b \cdot \frac{1}{b}\right) \]
\[ = \frac{b}{a} + \frac{4a}{b} + 1 \]

### Step 3: Set up the expression to maximize

We need to maximize:
\[ \frac{10}{5 + \alpha \cdot \beta} = \frac{10}{5 + \left(\frac{b}{a} + \frac{4a}{b} + 1\right)} \]

### Step 4: Simplify the expression

Substituting \( \alpha \cdot \beta \):
\[ = \frac{10}{6 + \frac{b}{a} + \frac{4a}{b}} \]

### Step 5: Find the minimum value of \( \alpha \cdot \beta \)

To maximize \( \frac{10}{5 + \alpha \cdot \beta} \), we need to minimize \( \alpha \cdot \beta \). We can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{\frac{b}{a} + \frac{4a}{b}}{2} \geq \sqrt{\frac{b}{a} \cdot \frac{4a}{b}} = 2 \]
Thus,
\[ \frac{b}{a} + \frac{4a}{b} \geq 4 \]

Adding 1 to both sides:
\[ \frac{b}{a} + \frac{4a}{b} + 1 \geq 5 \]

### Step 6: Substitute the minimum value into the expression

The minimum value of \( \alpha \cdot \beta \) is 5. Therefore:
\[ \frac{10}{5 + \alpha \cdot \beta} \leq \frac{10}{10} = 1 \]

### Conclusion

The maximum value of \( \frac{10}{5 + \alpha \cdot \beta} \) is:
\[ \boxed{1} \]