If ` x = ( sqrt( 3) + sqrt(2))/( sqrt(3) - sqrt(2)) and y = ( sqrt(3) - sqrt(2))/( sqrt(3) + sqrt(2))` then the value of ` ( x^(2) + 6 xy + y^(2))/( x^(2) - 6xy + y^(2))` is

Options

A. `(13)/(11)`
B. `(17)/(15)`
C. `(26)/(23)`
D. `(27)/(25)`

Answer

To solve the problem, we need to find the value of

\[ \frac{x^2 + 6xy + y^2}{x^2 - 6xy + y^2} \]

where

\[ x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \quad \text{and} \quad y = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}. \]

### Step 1: Rationalizing \( x \)

We start with \( x \):

\[ x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}. \]

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:

\[ x = \frac{(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}. \]

Calculating the numerator:

\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{6} = 5 + 2\sqrt{6}. \]

Calculating the denominator:

\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1. \]

Thus,

\[ x = 5 + 2\sqrt{6}. \]

### Step 2: Rationalizing \( y \)

Next, we rationalize \( y \):

\[ y = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}. \]

Again, multiply by the conjugate of the denominator:

\[ y = \frac{(\sqrt{3} - \sqrt{2})(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})}. \]

Calculating the numerator:

\[ (\sqrt{3} - \sqrt{2})^2 = 3 + 2 - 2\sqrt{6} = 5 - 2\sqrt{6}. \]

Calculating the denominator:

\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1. \]

Thus,

\[ y = 5 - 2\sqrt{6}. \]

### Step 3: Finding \( xy \)

Now we find \( xy \):

\[ xy = \left( \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \right) \left( \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \right) = 1. \]

### Step 4: Finding \( x^2 + y^2 \)

Next, we calculate \( x^2 + y^2 \):

\[ x^2 = (5 + 2\sqrt{6})^2 = 25 + 20\sqrt{6} + 24 = 49 + 20\sqrt{6}, \]

\[ y^2 = (5 - 2\sqrt{6})^2 = 25 - 20\sqrt{6} + 24 = 49 - 20\sqrt{6}. \]

Adding these:

\[ x^2 + y^2 = (49 + 20\sqrt{6}) + (49 - 20\sqrt{6}) = 98. \]

### Step 5: Finding \( x^2 + 6xy + y^2 \)

Now we calculate \( x^2 + 6xy + y^2 \):

\[ x^2 + 6xy + y^2 = x^2 + y^2 + 6 \cdot 1 = 98 + 6 = 104. \]

### Step 6: Finding \( x^2 - 6xy + y^2 \)

Next, we calculate \( x^2 - 6xy + y^2 \):

\[ x^2 - 6xy + y^2 = x^2 + y^2 - 6 \cdot 1 = 98 - 6 = 92. \]

### Step 7: Final Calculation

Now we can find the final value:

\[ \frac{x^2 + 6xy + y^2}{x^2 - 6xy + y^2} = \frac{104}{92} = \frac{26}{23}. \]

Thus, the value is

\[ \frac{26}{23}. \]