If a,b,c and A,B,C `in`R-{0} such that `aA+bB+cD+ sqrt((a^(2)+b^(2)+c^(2))(A^(2)+B^(2)+C^(2)))=0`, then value of `(aB)/(bA) +(bC)/(cB) + (cA)/(aC)` is

Options

A. 3
B. 4
C. 5
D. 6

Explanation

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

\[ aA + bB + cC + \sqrt{(a^2 + b^2 + c^2)(A^2 + B^2 + C^2)} = 0 \]

### Step 1: Rearranging the equation
We can rearrange the equation to isolate the square root term:

\[ aA + bB + cC = -\sqrt{(a^2 + b^2 + c^2)(A^2 + B^2 + C^2)} \]

### Step 2: Squaring both sides
Next, we square both sides to eliminate the square root:

\[ (aA + bB + cC)^2 = (a^2 + b^2 + c^2)(A^2 + B^2 + C^2) \]

### Step 3: Expanding both sides
Now we expand both sides. The left side expands to:

\[ a^2A^2 + b^2B^2 + c^2C^2 + 2(abAB + acAC + bcBC) \]

The right side remains:

\[ (a^2 + b^2 + c^2)(A^2 + B^2 + C^2) \]

### Step 4: Setting the expanded forms equal
We set the expanded forms equal to each other:

\[ a^2A^2 + b^2B^2 + c^2C^2 + 2(abAB + acAC + bcBC) = (a^2 + b^2 + c^2)(A^2 + B^2 + C^2) \]

### Step 5: Analyzing the equality
For the equality to hold, we can conclude that:

\[ abAB + acAC + bcBC = 0 \]

### Step 6: Finding the value of the expression
We need to find the value of:

\[ \frac{aB}{bA} + \frac{bC}{cB} + \frac{cA}{aC} \]

### Step 7: Using the equality derived
From the equality \( abAB + acAC + bcBC = 0 \), we can deduce that:

\[ \frac{aB}{bA} = k, \quad \frac{bC}{cB} = k, \quad \frac{cA}{aC} = k \]

This implies that all three fractions are equal to some constant \( k \).

### Step 8: Summing the fractions
Thus, we can write:

\[ \frac{aB}{bA} + \frac{bC}{cB} + \frac{cA}{aC} = k + k + k = 3k \]

### Step 9: Finding the value of \( k \)
Since \( abAB + acAC + bcBC = 0 \) implies that \( k = 1 \) (as each term must balance out to zero), we find:

\[ \frac{aB}{bA} + \frac{bC}{cB} + \frac{cA}{aC} = 3 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{3} \]