If `x=(sqrt3+sqrt2)/(sqrt3-sqrt2) and y=(sqrt3-sqrt2)/(sqrt3+sqrt2)`, then the value of `x^2+y^2` is

Options

A. 90
B. 98
C. 96
D. 94

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 \) 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: Rationalize \( x \)

To rationalize \( x \), we multiply the numerator and denominator by the conjugate of the denominator:

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

### Step 2: Simplify the denominator

The denominator simplifies as follows:

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

### Step 3: Simplify the numerator

Now simplify the numerator:

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

Thus, we have:

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

### Step 4: Rationalize \( y \)

Now we will rationalize \( y \):

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

### Step 5: Simplify the denominator for \( y \)

The denominator simplifies as follows:

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

### Step 6: Simplify the numerator for \( y \)

Now simplify the numerator:

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

Thus, we have:

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

### Step 7: Calculate \( x^2 \)

Now we calculate \( x^2 \):

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

### Step 8: Calculate \( y^2 \)

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

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

### Step 9: Add \( x^2 \) and \( y^2 \)

Now we add \( x^2 \) and \( y^2 \):

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

### Final Answer

Thus, the value of \( x^2 + y^2 \) is

\[ \boxed{98}. \]