If the roots of the equation
`(b-c) x^(2)+(c-a) x+(a-b) =0` be equal, then a,b,c are in

Options

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

Answer

To determine the relationship between \( a, b, c \) given that the roots of the quadratic equation

\[ (b-c)x^2 + (c-a)x + (a-b) = 0 \]

are equal, we will follow these steps:

### Step 1: Identify the coefficients
The coefficients of the quadratic equation are:
- \( A = b - c \) (coefficient of \( x^2 \))
- \( B = c - a \) (coefficient of \( x \))
- \( C = a - b \) (constant term)

### Step 2: Use the condition for equal roots
For a quadratic equation \( Ax^2 + Bx + C = 0 \) to have equal roots, the discriminant must be zero. The discriminant \( D \) is given by:

\[ D = B^2 - 4AC \]

Setting the discriminant to zero gives us:

\[ B^2 - 4AC = 0 \]

### Step 3: Substitute the coefficients into the discriminant
Substituting \( A, B, C \) into the discriminant:

\[ (c - a)^2 - 4(b - c)(a - b) = 0 \]

### Step 4: Expand and simplify the equation
Expanding the equation:

\[ (c - a)^2 = 4(b - c)(a - b) \]

Expanding both sides:

\[ (c^2 - 2ac + a^2) = 4[(b - c)(a - b)] \]

Expanding the right side:

\[ (b - c)(a - b) = ab - b^2 - ac + bc \]

Thus, we have:

\[ c^2 - 2ac + a^2 = 4(ab - b^2 - ac + bc) \]

### Step 5: Rearranging the equation
Rearranging gives:

\[ c^2 - 2ac + a^2 - 4ab + 4b^2 + 4ac - 4bc = 0 \]

Combining like terms:

\[ c^2 + 2ac + a^2 - 4ab + 4b^2 - 4bc = 0 \]

### Step 6: Analyze the condition
This equation can be interpreted in terms of the arithmetic progression (AP) of \( a, b, c \).

If \( a, b, c \) are in AP, then:

\[ b - a = c - b \implies 2b = a + c \]

This implies that:

\[ c - a = 2(b - a) \]

### Conclusion
Thus, if the roots of the quadratic equation are equal, it implies that \( a, b, c \) are in Arithmetic Progression (AP).