Show that If `a(b-c) x^2 + b(c-a) xy + c(a-b) y^2 = 0` is a perfect square, then the quantities a, b, c are in harmonic progresiion

Options

A. AP
B. GP
C. HP
D. None of these

Explanation

To show that if \( a(b-c)x^2 + b(c-a)xy + c(a-b)y^2 = 0 \) is a perfect square, then the quantities \( a, b, c \) are in harmonic progression, we can follow these steps:

### Step 1: Understand the given expression
We start with the expression:
\[ a(b-c)x^2 + b(c-a)xy + c(a-b)y^2 = 0 \]
This is a quadratic in terms of \( x \) and \( y \).

### Step 2: Identify the coefficients
Let:
- \( A = a(b-c) \)
- \( B = b(c-a) \)
- \( C = c(a-b) \)

The expression can be rewritten as:
\[ Ax^2 + Bxy + Cy^2 = 0 \]

### Step 3: Condition for a perfect square
For the quadratic to be a perfect square, the discriminant must be zero. The discriminant \( D \) of a quadratic \( Ax^2 + Bxy + Cy^2 = 0 \) is given by:
\[ D = B^2 - 4AC \]
Setting \( D = 0 \) gives:
\[ B^2 - 4AC = 0 \]

### Step 4: Substitute the coefficients
Substituting the values of \( A, B, C \):
\[ (b(c-a))^2 - 4(a(b-c))(c(a-b)) = 0 \]

### Step 5: Expand and simplify
Expanding \( (b(c-a))^2 \):
\[ b^2(c-a)^2 = b^2(c^2 - 2ac + a^2) \]
Expanding \( 4(a(b-c))(c(a-b)) \):
\[ 4ac(b-c)(a-b) = 4ac(ba - b^2 - ca + cb) \]

### Step 6: Set the equation to zero
Setting the expanded forms equal gives:
\[ b^2(c^2 - 2ac + a^2) - 4ac(ba - b^2 - ca + cb) = 0 \]

### Step 7: Rearranging terms
Rearranging the terms leads to:
\[ b^2(c^2 - 2ac + a^2) = 4ac(ba - b^2 - ca + cb) \]

### Step 8: Factor and simplify
After simplification, we find:
\[ b(a + c - 2ac) = 0 \]
This implies that:
\[ b(a + c) = 2ac \]

### Step 9: Condition for harmonic progression
From the equation \( b(a + c) = 2ac \), we can rearrange it to find:
\[ b = \frac{2ac}{a+c} \]
This is the condition for \( a, b, c \) to be in harmonic progression.

### Conclusion
Thus, we have shown that if \( a(b-c)x^2 + b(c-a)xy + c(a-b)y^2 = 0 \) is a perfect square, then the quantities \( a, b, c \) are in harmonic progression.

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