If a, b, c are in H.P., then `a^(2) (b - c)^(2), (b^(2))/(4) (c - a)^(2), c^(2) (a-b)^(2)` are in

Options

A. H.P.
B. G.P.
C. A.P.
D. none of these

Answer

To determine if the expressions \( a^2(b - c)^2 \), \( \frac{b^2}{4}(c - a)^2 \), and \( c^2(a - b)^2 \) are in Arithmetic Progression (A.P.) given that \( a, b, c \) are in Harmonic Progression (H.P.), we can follow these steps:

### Step 1: Understand the relationship between H.P. and A.P.
If \( a, b, c \) are in H.P., then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in A.P. This means:
\[ 2\frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
or equivalently,
\[ \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \]

### Step 2: Express \( b \) in terms of \( a \) and \( c \)
From the above relationship, we can express \( b \) in terms of \( a \) and \( c \):
\[ b = \frac{2ac}{a + c} \]

### Step 3: Substitute \( b \) into the expressions
Now we substitute \( b \) into the expressions \( a^2(b - c)^2 \), \( \frac{b^2}{4}(c - a)^2 \), and \( c^2(a - b)^2 \).

1. **First Expression**:
\[ a^2(b - c)^2 = a^2\left(\frac{2ac}{a+c} - c\right)^2 \]
Simplifying \( b - c \):
\[ b - c = \frac{2ac - c(a+c)}{a+c} = \frac{2ac - ac - c^2}{a+c} = \frac{ac - c^2}{a+c} = \frac{c(a - c)}{a+c} \]
Thus,
\[ a^2(b - c)^2 = a^2\left(\frac{c(a - c)}{a+c}\right)^2 = \frac{a^2c^2(a - c)^2}{(a+c)^2} \]

2. **Second Expression**:
\[ \frac{b^2}{4}(c - a)^2 = \frac{1}{4}\left(\frac{2ac}{a+c}\right)^2(c - a)^2 \]
Simplifying \( b^2 \):
\[ b^2 = \frac{4a^2c^2}{(a+c)^2} \]
Thus,
\[ \frac{b^2}{4}(c - a)^2 = \frac{a^2c^2(c - a)^2}{(a+c)^2} \]

3. **Third Expression**:
\[ c^2(a - b)^2 = c^2\left(a - \frac{2ac}{a+c}\right)^2 \]
Simplifying \( a - b \):
\[ a - b = a - \frac{2ac}{a+c} = \frac{a(a+c) - 2ac}{a+c} = \frac{a^2 - ac}{a+c} = \frac{a(a - c)}{a+c} \]
Thus,
\[ c^2(a - b)^2 = c^2\left(\frac{a(a - c)}{a+c}\right)^2 = \frac{c^2a^2(a - c)^2}{(a+c)^2} \]

### Step 4: Check if the three expressions are in A.P.
Now we have:
1. \( A = \frac{a^2c^2(a - c)^2}{(a+c)^2} \)
2. \( B = \frac{a^2c^2(c - a)^2}{(a+c)^2} \)
3. \( C = \frac{c^2a^2(a - c)^2}{(a+c)^2} \)

To check if \( A, B, C \) are in A.P., we need to verify:
\[ 2B = A + C \]

Substituting the values:
\[ 2 \cdot \frac{a^2c^2(c - a)^2}{(a+c)^2} = \frac{a^2c^2(a - c)^2}{(a+c)^2} + \frac{c^2a^2(a - c)^2}{(a+c)^2} \]

This simplifies to:
\[ \frac{2a^2c^2(c - a)^2}{(a+c)^2} = \frac{(a^2 + c^2)a^2c^2(a - c)^2}{(a+c)^2} \]
This confirms that the three expressions are indeed equal, thus they are in A.P.

### Conclusion
The three expressions \( a^2(b - c)^2 \), \( \frac{b^2}{4}(c - a)^2 \), and \( c^2(a - b)^2 \) are in Arithmetic Progression.

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