If both 'a' and 'b' are rational numbers, then 'a' and 'b' from `(3-sqrt(5))/(3+2sqrt(5))=asqrt(5)-b` , respectively are

Options

A. `(9)/(11),(19)/(11)`
B. `(19)/(11),(9)/(11)`
C. `(2)/(11),(8)/(11)`
D. `(10)/(11),(21)/(11)`

Answer

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

### Step 1: Rationalize the Denominator
To simplify the left-hand side, we will multiply the numerator and the denominator by the conjugate of the denominator, which is \(3 - 2\sqrt{5}\).

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

### Step 2: Simplify the Denominator
Now, we simplify the denominator using the difference of squares:

\[ (3 + 2\sqrt{5})(3 - 2\sqrt{5}) = 3^2 - (2\sqrt{5})^2 = 9 - 20 = -11 \]

### Step 3: Expand the Numerator
Next, we expand the numerator:

\[ (3 - \sqrt{5})(3 - 2\sqrt{5}) = 3 \cdot 3 - 3 \cdot 2\sqrt{5} - \sqrt{5} \cdot 3 + \sqrt{5} \cdot 2\sqrt{5} \]
\[ = 9 - 6\sqrt{5} - 3\sqrt{5} + 2 \cdot 5 = 9 - 9\sqrt{5} + 10 = 19 - 9\sqrt{5} \]

### Step 4: Combine the Results
Now, we can combine the results from the numerator and denominator:

\[ \frac{19 - 9\sqrt{5}}{-11} = \frac{-19 + 9\sqrt{5}}{11} = -\frac{19}{11} + \frac{9}{11}\sqrt{5} \]

### Step 5: Rearranging the Expression
We can rearrange this expression to match the form \(a\sqrt{5} - b\):

\[ \frac{9}{11}\sqrt{5} - \frac{19}{11} \]

### Step 6: Identify \(a\) and \(b\)
From the expression \(\frac{9}{11}\sqrt{5} - \frac{19}{11}\), we can identify:

\[ a = \frac{9}{11}, \quad b = \frac{19}{11} \]

### Conclusion
Thus, the values of \(a\) and \(b\) are:

\[ a = \frac{9}{11}, \quad b = \frac{19}{11} \]