Let `a = sqrt(6) - sqrt(5) , b = sqrt(5) - 2, c = 2 - sqrt(3)`
Then point out the correct alternative among the four alternatives gives bleow

Options

A. `b lt a lt c`
B. ` a lt c lt b`
C. ` b lt c lt a`
D. `b a lt b lt c`

Answer

To solve the problem, we need to analyze the expressions for \( a \), \( b \), and \( c \) given as:

- \( a = \sqrt{6} - \sqrt{5} \)
- \( b = \sqrt{5} - 2 \)
- \( c = 2 - \sqrt{3} \)

We will rationalize each expression to compare their values.

### Step 1: Rationalize \( a \)

We start with \( a = \sqrt{6} - \sqrt{5} \).

To rationalize, we multiply and divide by the conjugate:

\[ a = \frac{(\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5})}{\sqrt{6} + \sqrt{5}} = \frac{6 - 5}{\sqrt{6} + \sqrt{5}} = \frac{1}{\sqrt{6} + \sqrt{5}} \]

### Step 2: Rationalize \( b \)

Next, we rationalize \( b = \sqrt{5} - 2 \).

Again, we multiply and divide by the conjugate:

\[ b = \frac{(\sqrt{5} - 2)(\sqrt{5} + 2)}{\sqrt{5} + 2} = \frac{5 - 4}{\sqrt{5} + 2} = \frac{1}{\sqrt{5} + 2} \]

### Step 3: Rationalize \( c \)

Now, we rationalize \( c = 2 - \sqrt{3} \).

We multiply and divide by the conjugate:

\[ c = \frac{(2 - \sqrt{3})(2 + \sqrt{3})}{2 + \sqrt{3}} = \frac{4 - 3}{2 + \sqrt{3}} = \frac{1}{2 + \sqrt{3}} \]

### Step 4: Compare the values of \( a \), \( b \), and \( c \)

Now we have:

- \( a = \frac{1}{\sqrt{6} + \sqrt{5}} \)
- \( b = \frac{1}{\sqrt{5} + 2} \)
- \( c = \frac{1}{2 + \sqrt{3}} \)

To compare these fractions, we need to analyze their denominators:

1. **Denominator of \( a \)**: \( \sqrt{6} + \sqrt{5} \)
2. **Denominator of \( b \)**: \( \sqrt{5} + 2 \)
3. **Denominator of \( c \)**: \( 2 + \sqrt{3} \)

### Step 5: Determine the largest denominator

To find the smallest value among \( a \), \( b \), and \( c \), we need to find the largest denominator:

- **Estimate \( \sqrt{6} \approx 2.45 \) and \( \sqrt{5} \approx 2.24 \)**:
- \( \sqrt{6} + \sqrt{5} \approx 4.69 \)

- **Estimate \( \sqrt{5} \approx 2.24 \)**:
- \( \sqrt{5} + 2 \approx 4.24 \)

- **Estimate \( \sqrt{3} \approx 1.73 \)**:
- \( 2 + \sqrt{3} \approx 3.73 \)

### Conclusion

From the estimates, we see that:

- Denominator of \( c \) (approx. 3.73) is the smallest,
- Denominator of \( b \) (approx. 4.24) is larger,
- Denominator of \( a \) (approx. 4.69) is the largest.

Thus, the order of the values is:

\[ c < b < a \]

### Final Answer

The correct alternative is that \( c \) is the smallest, followed by \( b \), and then \( a \).

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