If a, b c are real and `a^(3)+b^(3)+c^(3) =3abc` and a+b+c `ne0`, then the relation between a, b,c will be

Options

A. a+b=c
B. a+c=b
C. a=b=c
D. b+c=a

Answer

To solve the problem, we start with the given equation:

\[ a^3 + b^3 + c^3 = 3abc \]

We also have the condition that \( a + b + c \neq 0 \).

### Step 1: Use the identity for the sum of cubes
We can use the identity that relates the sum of cubes to the product of sums:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

### Step 2: Set the equation to zero
Since we know from the problem statement that:

\[ a^3 + b^3 + c^3 = 3abc \]

We can substitute this into our identity:

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

### Step 3: Analyze the factors
The equation \( (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) = 0 \) gives us two cases to consider:

1. \( a + b + c = 0 \)
2. \( a^2 + b^2 + c^2 - ab - ac - bc = 0 \)

However, we are given that \( a + b + c \neq 0 \). Therefore, we must have:

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

### Step 4: Rearranging the equation
We can rearrange the equation:

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

### Step 5: Recognize the implications
The equation \( a^2 + b^2 + c^2 = ab + ac + bc \) can be rewritten as:

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

### Step 6: Conclude the relation
Since the sum of squares is equal to zero, each individual square must also be zero. This leads us to conclude:

\[ a - b = 0, \quad b - c = 0, \quad c - a = 0 \]

Thus, we have:

\[ a = b = c \]

### Final Relation
The relation between \( a, b, c \) is:

\[ a = b = c \]