If a, b, c are in AP, then `(a - c)^(2)` equals

Options

A. `4(b^(2) - ac)`
B. `4(b^(2) + ac)`
C. `4b^(2) - ac`
D. `b^(2) - 4ac`

Answer

To solve the problem, we need to show that if \( a, b, c \) are in Arithmetic Progression (AP), then \( (a - c)^2 \) equals \( 4(b^2 - ac) \).

### Step-by-Step Solution:

1. **Understanding the Condition of AP**:
Since \( a, b, c \) are in AP, we know that:
\[ b = \frac{a + c}{2} \]

2. **Expressing \( a + c \)**:
Rearranging the equation gives:
\[ a + c = 2b \]

3. **Finding \( (a - c)^2 \)**:
We can express \( (a - c)^2 \) using the identity:
\[ (a - c)^2 = a^2 - 2ac + c^2 \]

4. **Using the Square of a Sum**:
We can also express \( a^2 + c^2 \) using the square of the sum:
\[ a^2 + c^2 = (a + c)^2 - 2ac \]
Substituting \( a + c = 2b \):
\[ a^2 + c^2 = (2b)^2 - 2ac = 4b^2 - 2ac \]

5. **Substituting Back**:
Now substituting \( a^2 + c^2 \) back into the equation for \( (a - c)^2 \):
\[ (a - c)^2 = (a^2 + c^2) - 2ac = (4b^2 - 2ac) - 2ac \]
Simplifying this gives:
\[ (a - c)^2 = 4b^2 - 4ac \]

6. **Factoring**:
We can factor out a 4:
\[ (a - c)^2 = 4(b^2 - ac) \]

### Conclusion:
Thus, we have shown that:
\[ (a - c)^2 = 4(b^2 - ac) \]