If ` a^(2) + b^(2) + c^(2) - ab - bc - ca = 0 ` . Then a : b : c is

Options

A. ` 1 : 1 : 2 `
B. ` 1 : 1 : 1 `
C. ` 1 : 2 : 1 `
D. ` 2 : 1 : 1 `

Answer

To solve the equation \( a^2 + b^2 + c^2 - ab - bc - ca = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ a^2 + b^2 + c^2 - ab - bc - ca = 0 \]
We can rearrange it as:
\[ a^2 + b^2 + c^2 = ab + bc + ca \]

### Step 2: Analyze the equality condition
The equation \( a^2 + b^2 + c^2 = ab + bc + ca \) holds true under specific conditions. One of the conditions is when \( a = b = c \).

### Step 3: Substitute \( a = b = c \)
Let’s assume:
\[ a = b = c = k \]
for some constant \( k \). Substituting these values into the equation gives:
\[ k^2 + k^2 + k^2 = k \cdot k + k \cdot k + k \cdot k \]
This simplifies to:
\[ 3k^2 = 3k^2 \]
which is always true.

### Step 4: Determine the ratio
Since we have assumed \( a = b = c \), we can express the ratio \( a : b : c \) as:
\[ a : b : c = k : k : k = 1 : 1 : 1 \]

### Conclusion
Thus, the ratio \( a : b : c \) is:
\[ \boxed{1 : 1 : 1} \]

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