If `a ne b ne c`, then solution of equation
`|{:(x-a,x-b,x-c),(x-b,x-c,x-a),(x-c, x-a,x-b):}|=0" "is:`

Options

A. `x=0`
B. `x=a+b+c`
C. `x=1/2(a+b+c)`
D. `x=1/3(a+b+c)`

Answer

To solve the equation given by the determinant

\[ \left| \begin{array}{ccc} x-a & x-b & x-c \\ x-b & x-c & x-a \\ x-c & x-a & x-b \end{array} \right| = 0 \]

we will follow these steps:

### Step 1: Simplify the Determinant
We will add all three rows of the determinant together. This means we will replace the first row with the sum of all three rows.

\[ \text{New Row 1} = (x-a) + (x-b) + (x-c) = 3x - (a+b+c) \]

The determinant now looks like this:

\[ \left| \begin{array}{ccc} 3x - (a+b+c) & x-b & x-c \\ x-b & x-c & x-a \\ x-c & x-a & x-b \end{array} \right| = 0 \]

### Step 2: Factor Out Common Terms
Now we can factor out \(3x - (a+b+c)\) from the first row of the determinant:

\[ (3x - (a+b+c)) \left| \begin{array}{ccc} 1 & \frac{x-b}{3x - (a+b+c)} & \frac{x-c}{3x - (a+b+c)} \\ x-b & x-c & x-a \\ x-c & x-a & x-b \end{array} \right| = 0 \]

### Step 3: Set the Factors to Zero
The product of the two factors equals zero, which means either:

1. \(3x - (a+b+c) = 0\)
2. The determinant of the remaining 2x2 matrix equals zero.

We will focus on the first factor:

\[ 3x - (a+b+c) = 0 \]

### Step 4: Solve for x
Now, we can solve for \(x\):

\[ 3x = a + b + c \]

\[ x = \frac{a + b + c}{3} \]

### Conclusion
Thus, the solution to the given determinant equation is:

\[ x = \frac{a + b + c}{3} \]