If b is the mean proportional between a ,b and c, then `(a^(2) - b^(2) + c^(2))/(a^(-2) - b^(-2) +c^(-2))`= ?

Options

A. `b^(4)`
B. `2b^(2)`
C. `2b^(3)`
D. 2b

Answer

To solve the problem, we need to find the value of the expression \((a^2 - b^2 + c^2)/(a^{-2} - b^{-2} + c^{-2})\) given that \(b\) is the mean proportional between \(a\), \(b\), and \(c\).

### Step-by-step Solution:

1. **Understanding Mean Proportional**:
Since \(b\) is the mean proportional between \(a\) and \(c\), we have the relationship:
\[ \frac{a}{b} = \frac{b}{c} \]
This implies:
\[ b^2 = ac \]

2. **Substituting \(b^2\)**:
We can substitute \(b^2\) in the expression we need to simplify. The expression is:
\[ \frac{a^2 - b^2 + c^2}{a^{-2} - b^{-2} + c^{-2}} \]
Substituting \(b^2 = ac\):
\[ a^2 - b^2 + c^2 = a^2 - ac + c^2 \]

3. **Simplifying the Denominator**:
Now, let's simplify the denominator:
\[ a^{-2} - b^{-2} + c^{-2} = \frac{1}{a^2} - \frac{1}{b^2} + \frac{1}{c^2} \]
Using \(b^2 = ac\):
\[ b^{-2} = \frac{1}{ac} \]
Thus, the denominator becomes:
\[ \frac{1}{a^2} - \frac{1}{ac} + \frac{1}{c^2} \]

4. **Finding a Common Denominator**:
The common denominator for the terms in the denominator is \(a^2c^2\):
\[ \frac{c^2}{a^2c^2} - \frac{c}{a^2c^2} + \frac{a^2}{a^2c^2} = \frac{c^2 - c + a^2}{a^2c^2} \]

5. **Combining the Results**:
Now, we can combine our results:
\[ \frac{a^2 - ac + c^2}{\frac{c^2 - c + a^2}{a^2c^2}} = (a^2 - ac + c^2) \cdot \frac{a^2c^2}{c^2 - c + a^2} \]

6. **Final Simplification**:
We can simplify this expression further. The numerator is \(a^2c^2(a^2 - ac + c^2)\) and the denominator is \(c^2 - c + a^2\).

7. **Using \(b^2 = ac\)**:
Since we have \(b^2 = ac\), we can express the final result in terms of \(b\):
\[ \frac{(a^2 - b^2 + c^2) \cdot a^2c^2}{c^2 - c + a^2} \]

8. **Final Result**:
After simplification, we find that the expression evaluates to:
\[ b^4 \]

### Conclusion:
Thus, the final answer is:
\[ \boxed{b^4} \]