If `(sqrt(a+2b)+sqrt(a-2b))/(sqrt(a+2b)-sqrt(a-2b))=sqrt(3)`, then a : b is equal to

Options

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

Answer

To solve the equation

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

we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fraction:

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

### Step 2: Expand the Right Side
Expanding the right side gives us:

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

### Step 3: Rearranging the Equation
Rearranging the equation leads to:

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

### Step 4: Combine Like Terms
Combining like terms, we have:

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

### Step 5: Factor Out Common Terms
Factoring out common terms gives us:

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

### Step 6: Square Both Sides
Now, we square both sides to eliminate the square roots:

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

### Step 7: Expand Both Sides
Expanding both sides results in:

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

This simplifies to:

\[ (4 - 2\sqrt{3})(a + 2b) = (4 + 2\sqrt{3})(a - 2b). \]

### Step 8: Distribute
Distributing gives:

\[ (4 - 2\sqrt{3})a + (8 - 4\sqrt{3})b = (4 + 2\sqrt{3})a - (8 + 4\sqrt{3})b. \]

### Step 9: Collect Like Terms
Collecting like terms leads to:

\[ (4 - 2\sqrt{3})a - (4 + 2\sqrt{3})a = -(8 + 4\sqrt{3})b - (8 - 4\sqrt{3})b. \]

### Step 10: Simplify
This simplifies to:

\[ (-4\sqrt{3})a = -16b. \]

### Step 11: Solve for a : b
Dividing both sides by -4 gives:

\[ \sqrt{3}a = 4b. \]

Thus, we have:

\[ \frac{a}{b} = \frac{4}{\sqrt{3}}. \]

### Final Ratio
Therefore, the ratio \( a : b \) is:

\[ a : b = 4 : \sqrt{3}. \]

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