If the arithmetic mean between a and b equals n times their geometric mean, then find the ratio a : b.

Options

A. `(2n^2 + 1) pm 2n sqrt(n^2 -1)`
B. `(2n^2 - 1) pm 2n sqrt(n^2 - 1)`
C. `(2n^2 - 1) pm 2n sqrt(n^2 + 1)`
D. none of these

Answer

To solve the problem, we need to find the ratio \( a : b \) given that the arithmetic mean of \( a \) and \( b \) equals \( n \) times their geometric mean.

### Step-by-Step Solution:

1. **Write the expressions for the arithmetic mean and geometric mean:**
- The arithmetic mean (AM) of \( a \) and \( b \) is given by:
\[ AM = \frac{a + b}{2} \]
- The geometric mean (GM) of \( a \) and \( b \) is given by:
\[ GM = \sqrt{ab} \]

2. **Set up the equation based on the problem statement:**
According to the problem, the arithmetic mean equals \( n \) times the geometric mean:
\[ \frac{a + b}{2} = n \sqrt{ab} \]

3. **Multiply both sides by 2 to eliminate the fraction:**
\[ a + b = 2n \sqrt{ab} \]

4. **Square both sides to eliminate the square root:**
\[ (a + b)^2 = (2n \sqrt{ab})^2 \]
This simplifies to:
\[ a^2 + 2ab + b^2 = 4n^2 ab \]

5. **Rearrange the equation:**
Move all terms to one side:
\[ a^2 + b^2 + 2ab - 4n^2 ab = 0 \]
This simplifies to:
\[ a^2 + b^2 + (2 - 4n^2)ab = 0 \]

6. **Assume the ratio \( k = \frac{a}{b} \):**
We can express \( a \) in terms of \( b \):
\[ a = kb \]
Substitute this into the equation:
\[ (kb)^2 + b^2 + (2 - 4n^2)kb^2 = 0 \]
This simplifies to:
\[ k^2b^2 + b^2 + (2 - 4n^2)kb^2 = 0 \]

7. **Factor out \( b^2 \):**
\[ b^2(k^2 + 1 + (2 - 4n^2)k) = 0 \]
Since \( b^2 \neq 0 \), we can simplify to:
\[ k^2 + (2 - 4n^2)k + 1 = 0 \]

8. **Use the quadratic formula to solve for \( k \):**
The quadratic formula is given by:
\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = (2 - 4n^2), c = 1 \):
\[ k = \frac{-(2 - 4n^2) \pm \sqrt{(2 - 4n^2)^2 - 4}}{2} \]

9. **Simplify the expression under the square root:**
\[ k = \frac{-(2 - 4n^2) \pm \sqrt{(2 - 4n^2)^2 - 4}}{2} \]
This will give us the values of \( k \).

10. **Find the ratio \( a : b \):**
The ratio \( a : b \) is given by \( k : 1 \).

### Final Result:
The ratio \( a : b \) can be expressed in terms of \( n \) as:
\[ a : b = k : 1 \]