If a, b, c, d are four non - zero real numbers such that `(d+a-b)^(2)+(d+b-c)^(2)=0` and roots of the equation `a(b-c)x^(2)+b(c-a)x+c(a-b)=0` are real equal, then

Options

A. `a+b+c ne 0`
B. a, b, c are in H.P.
C. a, b, c are in A.P.
D. a, b, c are in G.P.

Answer

To solve the given problem step by step, we will analyze the conditions provided and derive the necessary relationships.

### Step 1: Analyze the first condition
We are given that:
\[ (d + a - b)^2 + (d + b - c)^2 = 0 \]
Since both terms are squares, for their sum to be zero, each term must independently be zero:
1. \(d + a - b = 0\)
2. \(d + b - c = 0\)

### Step 2: Solve the equations
From the first equation:
\[ d + a - b = 0 \implies d = b - a \]
From the second equation:
\[ d + b - c = 0 \implies d = c - b \]

### Step 3: Set the two expressions for \(d\) equal
Since both expressions for \(d\) are equal, we have:
\[ b - a = c - b \]
Rearranging gives:
\[ b - a + b = c \implies 2b = a + c \implies a + c = 2b \]
This indicates that \(a\), \(b\), and \(c\) are in Arithmetic Progression (AP).

### Step 4: Analyze the second condition
Next, we need to check the condition for the quadratic equation:
\[ a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \]
For the roots to be real and equal, the discriminant must be zero.

### Step 5: Calculate the discriminant
The discriminant \(D\) of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by:
\[ D = B^2 - 4AC \]
Here, \(A = a(b - c)\), \(B = b(c - a)\), and \(C = c(a - b)\). Thus:
\[ D = [b(c - a)]^2 - 4[a(b - c)][c(a - b)] \]

### Step 6: Set the discriminant to zero
Setting the discriminant to zero gives:
\[ [b(c - a)]^2 - 4[a(b - c)][c(a - b)] = 0 \]

### Step 7: Simplify the equation
We can analyze the terms:
1. \(b(c - a) = b(c - a)\)
2. \(4[a(b - c)][c(a - b)]\)

This leads us to the condition that needs to be satisfied for the roots to be equal.

### Step 8: Conclusion
From the analysis, we have established that:
- \(a + c = 2b\) implies \(a, b, c\) are in AP.
- The condition derived from the discriminant must also hold.

Thus, the final conclusion is that \(a, b, c\) are in Arithmetic Progression (AP).