In a right angled triangle the sides are a,b and c with c hypotenuse and `x-b ne 1`, `c+b ne 1`. Then the value of `(log_(c+b)a+log_(c-b)a)//2log_(c+b)axxlog_(c-a)` a will be

Options

A. `2`
B. `(1)/(2)`
C. `-1`
D. `1`

Answer

To solve the problem, we need to evaluate the expression given in the question step by step. The expression is:

\[ \frac{\log_{(c+b)} a + \log_{(c-b)} a}{2 \log_{(c+b)} a \cdot \log_{(c-a)} a} \]

### Step 1: Use the Change of Base Formula
We can use the change of base formula for logarithms, which states that:

\[ \log_b a = \frac{\log_k a}{\log_k b} \]

for any base \( k \). We will use base 10 (or natural logarithm) for our calculations.

Applying the change of base formula:

\[ \log_{(c+b)} a = \frac{\log a}{\log (c+b)} \]
\[ \log_{(c-b)} a = \frac{\log a}{\log (c-b)} \]

### Step 2: Substitute into the Expression
Substituting these into the expression gives:

\[ \frac{\frac{\log a}{\log (c+b)} + \frac{\log a}{\log (c-b)}}{2 \cdot \frac{\log a}{\log (c+b)} \cdot \frac{\log a}{\log (c-a)}} \]

### Step 3: Simplify the Numerator
The numerator becomes:

\[ \frac{\log a}{\log (c+b)} + \frac{\log a}{\log (c-b)} = \log a \left( \frac{1}{\log (c+b)} + \frac{1}{\log (c-b)} \right) \]

### Step 4: Simplify the Denominator
The denominator becomes:

\[ 2 \cdot \frac{\log a}{\log (c+b)} \cdot \frac{\log a}{\log (c-a)} = 2 \cdot \frac{(\log a)^2}{\log (c+b) \cdot \log (c-a)} \]

### Step 5: Combine the Expression
Now we can combine the simplified numerator and denominator:

\[ \frac{\log a \left( \frac{1}{\log (c+b)} + \frac{1}{\log (c-b)} \right)}{2 \cdot \frac{(\log a)^2}{\log (c+b) \cdot \log (c-a)}} \]

### Step 6: Further Simplification
This can be simplified to:

\[ \frac{\log a \cdot \log (c+b) \cdot \log (c-a)}{2 \cdot (\log a)^2} \left( \frac{1}{\log (c+b)} + \frac{1}{\log (c-b)} \right) \]

### Step 7: Cancel Out \(\log a\)
Cancelling one \(\log a\) from the numerator and denominator gives:

\[ \frac{\log (c+b) \cdot \log (c-a)}{2 \cdot \log a} \left( \frac{1}{\log (c+b)} + \frac{1}{\log (c-b)} \right) \]

### Step 8: Final Expression
This can be further simplified to obtain the final expression, but since the question asks for the value of \( a \), we can conclude that the expression simplifies to a constant value depending on the parameters \( c \) and \( b \).

### Conclusion
Thus, the value of \( a \) will depend on the specific values of \( b \) and \( c \) in the triangle.