If `a + b = 2c,` then the value `(a)/(a -c) + (c)/( b -c)` is equal to (where ` a ne b ne c`)

Options

A. `-1`
B. 1
C. 0
D. `1/2`

Answer

To solve the problem, we need to find the value of the expression \(\frac{a}{a - c} + \frac{c}{b - c}\) given that \(a + b = 2c\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ a + b = 2c \]

2. **Rearrange the equation to express \(b\) in terms of \(a\) and \(c\):**
\[ b = 2c - a \]

3. **Substitute \(b\) into the expression:**
We need to find:
\[ \frac{a}{a - c} + \frac{c}{b - c} \]
Substitute \(b = 2c - a\) into the second term:
\[ \frac{c}{(2c - a) - c} = \frac{c}{c - a} \]

4. **Rewrite the expression:**
Now we have:
\[ \frac{a}{a - c} + \frac{c}{c - a} \]

5. **Find a common denominator:**
The common denominator for the two fractions is \((a - c)(c - a)\). Notice that \(c - a = -(a - c)\), so we can rewrite the expression:
\[ \frac{a(c - a) + c(a - c)}{(a - c)(c - a)} \]
This simplifies to:
\[ \frac{a(c - a) - c(a - c)}{(a - c)(c - a)} \]

6. **Simplify the numerator:**
Expanding the numerator:
\[ ac - a^2 - ac + c^2 = c^2 - a^2 \]
Thus, we have:
\[ \frac{c^2 - a^2}{(a - c)(c - a)} = \frac{(c - a)(c + a)}{(a - c)(c - a)} \]

7. **Cancel out the common terms:**
Since \((c - a) = -(a - c)\), we can cancel:
\[ \frac{(c + a)}{-1} = -(c + a) \]

8. **Final result:**
Since \(a + b = 2c\), we can replace \(c + a\) with \(b\):
\[ -(c + a) = -b \]

Thus, the value of the expression \(\frac{a}{a - c} + \frac{c}{b - c}\) is equal to \(-b\).