Let ` x = ( sqrt(13) + sqrt(11))/( sqrt(13) - sqrt(11))`and y ` = (1)/(x)` then the value of `3 x^(2) - 5x y + 3y ^(2)` is

Options

A. 1717
B. 1177
C. 1771
D. 1171

Answer

To solve the problem, we need to find the value of the expression \(3x^2 - 5xy + 3y^2\) given \(x = \frac{\sqrt{13} + \sqrt{11}}{\sqrt{13} - \sqrt{11}}\) and \(y = \frac{1}{x}\).

### Step 1: Calculate \(x\)
We start with the expression for \(x\):
\[ x = \frac{\sqrt{13} + \sqrt{11}}{\sqrt{13} - \sqrt{11}} \]
To simplify \(x\), we can multiply the numerator and denominator by the conjugate of the denominator:
\[ x = \frac{(\sqrt{13} + \sqrt{11})(\sqrt{13} + \sqrt{11})}{(\sqrt{13} - \sqrt{11})(\sqrt{13} + \sqrt{11})} \]
Calculating the numerator:
\[ (\sqrt{13} + \sqrt{11})^2 = 13 + 11 + 2\sqrt{13 \cdot 11} = 24 + 2\sqrt{143} \]
Calculating the denominator using the difference of squares:
\[ (\sqrt{13})^2 - (\sqrt{11})^2 = 13 - 11 = 2 \]
Thus, we have:
\[ x = \frac{24 + 2\sqrt{143}}{2} = 12 + \sqrt{143} \]

### Step 2: Calculate \(y\)
Now we calculate \(y\):
\[ y = \frac{1}{x} = \frac{1}{12 + \sqrt{143}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate:
\[ y = \frac{12 - \sqrt{143}}{(12 + \sqrt{143})(12 - \sqrt{143})} \]
Calculating the denominator:
\[ (12)^2 - (\sqrt{143})^2 = 144 - 143 = 1 \]
Thus, we have:
\[ y = 12 - \sqrt{143} \]

### Step 3: Substitute \(x\) and \(y\) into the expression
Now we substitute \(x\) and \(y\) into the expression \(3x^2 - 5xy + 3y^2\):
1. Calculate \(x^2\):
\[ x^2 = (12 + \sqrt{143})^2 = 144 + 24\sqrt{143} + 143 = 287 + 24\sqrt{143} \]
2. Calculate \(y^2\):
\[ y^2 = (12 - \sqrt{143})^2 = 144 - 24\sqrt{143} + 143 = 287 - 24\sqrt{143} \]
3. Calculate \(xy\):
\[ xy = (12 + \sqrt{143})(12 - \sqrt{143}) = 144 - 143 = 1 \]

### Step 4: Substitute into the expression
Now substitute \(x^2\), \(y^2\), and \(xy\) into the expression:
\[ 3x^2 = 3(287 + 24\sqrt{143}) = 861 + 72\sqrt{143} \]
\[ 3y^2 = 3(287 - 24\sqrt{143}) = 861 - 72\sqrt{143} \]
\[ -5xy = -5(1) = -5 \]
Putting it all together:
\[ 3x^2 - 5xy + 3y^2 = (861 + 72\sqrt{143}) + (-5) + (861 - 72\sqrt{143}) \]
The terms with \(\sqrt{143}\) cancel out:
\[ = 861 + 861 - 5 = 1717 \]

### Final Answer
The value of \(3x^2 - 5xy + 3y^2\) is:
\[ \boxed{1717} \]