If `4/(2+sqrt3+sqrt7)=sqrta+sqrtb-sqrtc`,then which of the following can be ture -

Options

A. a=1, b=4/3, c=7/3
B. `a=1, b=2/3, c=7/9
C. a=2/3, b=1, c=7/3
D. a= 7/9, b=4/3, c=1

Answer

To solve the equation \( \frac{4}{2 + \sqrt{3} + \sqrt{7}} = \sqrt{a} + \sqrt{b} - \sqrt{c} \), we will manipulate the left-hand side to match the right-hand side.

### Step-by-step Solution:

1. **Rationalize the Denominator**:
We start with the expression \( \frac{4}{2 + \sqrt{3} + \sqrt{7}} \). To simplify this, we can multiply the numerator and the denominator by the conjugate of the denominator, which is \( 2 + \sqrt{3} - \sqrt{7} \).

\[ \frac{4(2 + \sqrt{3} - \sqrt{7})}{(2 + \sqrt{3} + \sqrt{7})(2 + \sqrt{3} - \sqrt{7})} \]

2. **Calculate the Denominator**:
The denominator can be simplified using the difference of squares formula:

\[ (2 + \sqrt{3})^2 - (\sqrt{7})^2 \]

First, calculate \( (2 + \sqrt{3})^2 \):

\[ (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

Now, calculate \( \sqrt{7}^2 = 7 \). Therefore, the denominator becomes:

\[ (7 + 4\sqrt{3}) - 7 = 4\sqrt{3} \]

3. **Simplify the Expression**:
Now we can substitute back into our expression:

\[ \frac{4(2 + \sqrt{3} - \sqrt{7})}{4\sqrt{3}} = \frac{2 + \sqrt{3} - \sqrt{7}}{\sqrt{3}} \]

This can be separated into individual fractions:

\[ \frac{2}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{3}} - \frac{\sqrt{7}}{\sqrt{3}} = \frac{2}{\sqrt{3}} + 1 - \frac{\sqrt{7}}{\sqrt{3}} \]

4. **Combine the Terms**:
We can express the terms as:

\[ 1 + \frac{2 - \sqrt{7}}{\sqrt{3}} \]

To express \( \frac{2 - \sqrt{7}}{\sqrt{3}} \) in a more manageable form, we can rewrite it as:

\[ \sqrt{\frac{4}{3}} - \sqrt{\frac{7}{3}} \]

5. **Final Form**:
Thus, we can express the entire left-hand side as:

\[ \sqrt{\frac{4}{3}} + 1 - \sqrt{\frac{7}{3}} \]

This matches the form \( \sqrt{a} + \sqrt{b} - \sqrt{c} \) where:
- \( a = \frac{4}{3} \)
- \( b = 1 \)
- \( c = \frac{7}{3} \)

### Conclusion:
From the above steps, we can conclude that the values of \( a \), \( b \), and \( c \) are:
- \( a = \frac{4}{3} \)
- \( b = 1 \)
- \( c = \frac{7}{3} \)

Thus, the correct option that matches is the first option.