If ` ( 2 + sqrt(3)) a = (2 - sqrt(3)) b = 1 `then the value of `(1)/(a) + (1)/( b)` is

Options

A. A)1
B. B)2
C. C)`2 sqrt(3)`
D. D)4

Answer

To solve the problem, we start with the given equations:

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

From this, we can express \(a\) and \(b\) in terms of the constant 1.

### Step 1: Express \(a\) in terms of the constant
From the equation \((2 + \sqrt{3})a = 1\), we can isolate \(a\):

\[ a = \frac{1}{2 + \sqrt{3}} \]

### Step 2: Express \(b\) in terms of the constant
Similarly, from the equation \((2 - \sqrt{3})b = 1\), we isolate \(b\):

\[ b = \frac{1}{2 - \sqrt{3}} \]

### Step 3: Find \(\frac{1}{a}\) and \(\frac{1}{b}\)
Now we need to find \(\frac{1}{a}\) and \(\frac{1}{b}\):

\[ \frac{1}{a} = 2 + \sqrt{3} \]
\[ \frac{1}{b} = 2 - \sqrt{3} \]

### Step 4: Add \(\frac{1}{a}\) and \(\frac{1}{b}\)
Now we can add these two fractions:

\[ \frac{1}{a} + \frac{1}{b} = (2 + \sqrt{3}) + (2 - \sqrt{3}) \]

### Step 5: Simplify the expression
When we add these two expressions, the \(\sqrt{3}\) terms cancel out:

\[ \frac{1}{a} + \frac{1}{b} = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \]

### Final Answer
Thus, the value of \(\frac{1}{a} + \frac{1}{b}\) is:

\[ \frac{1}{a} + \frac{1}{b} = 4 \]