If `(7+sqrt(5))/(7-sqrt(5))-(7-sqrt(5))/(7+sqrt(5))=a+7sqrt(5)b`, determine the rational number.

Options

A. `a=-2`, `b=(2)/(11)`
B. `a=0`, `b=(1)/(11)`
C. `a=-1`, `b=(1)/(11)`
D. `a=-2`, `b=-11`

Answer

To solve the equation

\[ \frac{7 + \sqrt{5}}{7 - \sqrt{5}} - \frac{7 - \sqrt{5}}{7 + \sqrt{5}} = a + 7\sqrt{5}b, \]

we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((7 - \sqrt{5})(7 + \sqrt{5})\).

### Step 2: Rewrite the fractions
Rewriting the fractions with the common denominator, we have:

\[ \frac{(7 + \sqrt{5})^2 - (7 - \sqrt{5})^2}{(7 - \sqrt{5})(7 + \sqrt{5})}. \]

### Step 3: Expand the numerators
Using the formula for the square of a binomial, we expand:

- \((7 + \sqrt{5})^2 = 49 + 14\sqrt{5} + 5 = 54 + 14\sqrt{5}\)
- \((7 - \sqrt{5})^2 = 49 - 14\sqrt{5} + 5 = 54 - 14\sqrt{5}\)

Now substituting back into the equation gives:

\[ \frac{(54 + 14\sqrt{5}) - (54 - 14\sqrt{5})}{(7 - \sqrt{5})(7 + \sqrt{5})}. \]

### Step 4: Simplify the numerator
The numerator simplifies to:

\[ 54 + 14\sqrt{5} - 54 + 14\sqrt{5} = 28\sqrt{5}. \]

### Step 5: Simplify the denominator
The denominator simplifies to:

\[ (7 - \sqrt{5})(7 + \sqrt{5}) = 49 - 5 = 44. \]

### Step 6: Combine the results
Now we have:

\[ \frac{28\sqrt{5}}{44} = \frac{7\sqrt{5}}{11}. \]

### Step 7: Set the equation
Now we can equate this to \(a + 7\sqrt{5}b\):

\[ \frac{7\sqrt{5}}{11} = a + 7\sqrt{5}b. \]

### Step 8: Identify \(a\) and \(b\)
From the equation, we can see:

- The rational part \(a = 0\)
- The coefficient of \(\sqrt{5}\) gives us \(7b = \frac{7}{11}\), thus \(b = \frac{1}{11}\).

### Final Answer
The rational number \(a\) is:

\[ \boxed{0}. \]