If `a, b, c` are in HP, then `(a - b)/( b- c)` is equal to

Options

A. `(a)/(b)`
B. `(b)/(a)`
C. `(a)/(c)`
D. `(c)/(b)`

Answer

To solve the problem, we need to find the value of \((a - b)/(b - c)\) given that \(a, b, c\) are in Harmonic Progression (HP).

### Step-by-step Solution:

1. **Understanding Harmonic Progression (HP)**:
- If \(a, b, c\) are in HP, then the reciprocals \(1/a, 1/b, 1/c\) are in Arithmetic Progression (AP).
- The condition for \(b\) in terms of \(a\) and \(c\) in HP is given by:
\[ 2/b = 1/a + 1/c \]
- Rearranging this gives:
\[ b = \frac{2ac}{a + c} \]

2. **Finding \(a - b\)**:
- Substitute the value of \(b\) into \(a - b\):
\[ a - b = a - \frac{2ac}{a + c} \]
- To combine these, we can express \(a\) with a common denominator:
\[ a - b = \frac{a(a + c) - 2ac}{a + c} = \frac{a^2 + ac - 2ac}{a + c} = \frac{a^2 - ac}{a + c} \]

3. **Finding \(b - c\)**:
- Now, we need to find \(b - c\):
\[ b - c = \frac{2ac}{a + c} - c \]
- Again, express \(c\) with a common denominator:
\[ b - c = \frac{2ac - c(a + c)}{a + c} = \frac{2ac - ac - c^2}{a + c} = \frac{ac - c^2}{a + c} \]

4. **Calculating \((a - b)/(b - c)\)**:
- Now we can substitute our expressions for \(a - b\) and \(b - c\):
\[ \frac{a - b}{b - c} = \frac{\frac{a^2 - ac}{a + c}}{\frac{ac - c^2}{a + c}} = \frac{a^2 - ac}{ac - c^2} \]

5. **Simplifying the Expression**:
- We can factor the numerator and denominator:
\[ = \frac{a(a - c)}{c(a - c)} \]
- If \(a \neq c\), we can cancel \(a - c\):
\[ = \frac{a}{c} \]

### Final Result:
Thus, we find that:
\[ \frac{a - b}{b - c} = \frac{a}{c} \]