If x,y are rational numbers and `(5+sqrt(11))/(3-2sqrt(11))=x+ysqrt(11)`.
The values of x and y are

Options

A. `x=(-14)/(17)`, `y=(-13)/(26)`
B. `x=(4)/(13)`, `y=(11)/(17)`
C. `x=(-27)/(25)`, `y=(-11)/(37)`
D. `x=(-37)/(35)`, `y=(-13)/(35)`

Answer

To solve the equation \(\frac{5 + \sqrt{11}}{3 - 2\sqrt{11}} = x + y\sqrt{11}\), where \(x\) and \(y\) are rational numbers, we will follow these steps:

### Step 1: Rationalize the Denominator
To eliminate the square root from the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \(3 + 2\sqrt{11}\).

\[ \frac{(5 + \sqrt{11})(3 + 2\sqrt{11})}{(3 - 2\sqrt{11})(3 + 2\sqrt{11})} \]

### Step 2: Calculate the Denominator
Using the difference of squares formula \(a^2 - b^2\):

\[ (3 - 2\sqrt{11})(3 + 2\sqrt{11}) = 3^2 - (2\sqrt{11})^2 = 9 - 4 \cdot 11 = 9 - 44 = -35 \]

### Step 3: Calculate the Numerator
Now we expand the numerator:

\[ (5 + \sqrt{11})(3 + 2\sqrt{11}) = 5 \cdot 3 + 5 \cdot 2\sqrt{11} + \sqrt{11} \cdot 3 + \sqrt{11} \cdot 2\sqrt{11} \]
\[ = 15 + 10\sqrt{11} + 3\sqrt{11} + 2 \cdot 11 \]
\[ = 15 + 10\sqrt{11} + 3\sqrt{11} + 22 = 37 + 13\sqrt{11} \]

### Step 4: Combine the Results
Now we can write the expression as:

\[ \frac{37 + 13\sqrt{11}}{-35} \]

This can be separated into two parts:

\[ = \frac{37}{-35} + \frac{13\sqrt{11}}{-35} = -\frac{37}{35} - \frac{13}{35}\sqrt{11} \]

### Step 5: Identify \(x\) and \(y\)
Now we can compare this with \(x + y\sqrt{11}\):

\[ x = -\frac{37}{35}, \quad y = -\frac{13}{35} \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:

\[ x = -\frac{37}{35}, \quad y = -\frac{13}{35} \]

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