If the expression `a^(2)(b^(2)-c^(2))x^(2)+b^(2)(c^(2)-a^(2))x+c^(2)(a^(2)-b^(2))` is a perfect square, then

Options

A. a, b, c are in A.P.
B. `a^(2), b^(2), c^(2)` are in A.P.
C. `a^(2), b^(2), c^(2)` are in H.P.
D. `a^(2), b^(2), c^(2)` are in G.P.

Explanation

To determine the conditions under which the expression \( a^2(b^2 - c^2)x^2 + b^2(c^2 - a^2)x + c^2(a^2 - b^2) \) is a perfect square, we will follow these steps:

### Step 1: Define the Expression
Let \( f(x) = a^2(b^2 - c^2)x^2 + b^2(c^2 - a^2)x + c^2(a^2 - b^2) \).

### Step 2: Identify the Coefficients
The coefficients of the quadratic expression are:
- Coefficient of \( x^2 \): \( a^2(b^2 - c^2) \)
- Coefficient of \( x \): \( b^2(c^2 - a^2) \)
- Constant term: \( c^2(a^2 - b^2) \)

### Step 3: Sum of Coefficients
To check if the expression can be a perfect square, we calculate the sum of the coefficients by substituting \( x = 1 \):
\[ f(1) = a^2(b^2 - c^2) + b^2(c^2 - a^2) + c^2(a^2 - b^2) \]
Simplifying this:
\[ = a^2b^2 - a^2c^2 + b^2c^2 - b^2a^2 + c^2a^2 - c^2b^2 \]
Notice that the terms cancel out:
\[ = 0 \]

### Step 4: Roots of the Quadratic
Since the sum of the coefficients is zero, \( x = 1 \) is one of the roots of the quadratic. For the expression to be a perfect square, the other root must also be \( x = 1 \).

### Step 5: Product of Roots
The product of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \frac{c}{a} \). Here, it translates to:
\[ \text{Product of roots} = \frac{c^2(a^2 - b^2)}{a^2(b^2 - c^2)} \]
Setting this equal to \( 1 \) (since both roots are \( 1 \)):
\[ \frac{c^2(a^2 - b^2)}{a^2(b^2 - c^2)} = 1 \]

### Step 6: Cross Multiply
Cross multiplying gives:
\[ c^2(a^2 - b^2) = a^2(b^2 - c^2) \]
Expanding both sides:
\[ c^2a^2 - c^2b^2 = a^2b^2 - a^2c^2 \]
Rearranging terms leads to:
\[ c^2a^2 + a^2c^2 = a^2b^2 + c^2b^2 \]
This simplifies to:
\[ b^2(a^2 + c^2) = 2a^2c^2 \]

### Step 7: Conclusion
This implies that:
\[ \frac{1}{c^2} + \frac{1}{a^2} = \frac{2}{b^2} \]
Thus, \( a^2, b^2, c^2 \) are in Harmonic Progression (HP).

### Final Answer
The condition for the expression to be a perfect square is that \( a^2, b^2, c^2 \) are in Harmonic Progression.

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