If ` x = a ( b - c) , y = b ( c - a) z = c ( a - b)` then the value of `(( x)/( a))^(3) + ((y)/( b))^(3) + ((z)/( c))^(3)` is

Options

A. `(2xyz)/( abc)`
B. `(xyz)/(abc)`
C. 0
D. `(3xyz)/(abc)`

Answer

To solve the problem, we will follow these steps:

Given:
- \( x = a(b - c) \)
- \( y = b(c - a) \)
- \( z = c(a - b) \)

We need to find the value of:
\[ \left(\frac{x}{a}\right)^3 + \left(\frac{y}{b}\right)^3 + \left(\frac{z}{c}\right)^3 \]

### Step 1: Substitute the values of x, y, and z into the expression
We start by substituting the values of \(x\), \(y\), and \(z\) into the expression:
\[ \frac{x}{a} = \frac{a(b - c)}{a} = b - c \]
\[ \frac{y}{b} = \frac{b(c - a)}{b} = c - a \]
\[ \frac{z}{c} = \frac{c(a - b)}{c} = a - b \]

### Step 2: Rewrite the expression
Now we can rewrite the expression:
\[ (b - c)^3 + (c - a)^3 + (a - b)^3 \]

### Step 3: Use the identity for the sum of cubes
We can use the identity:
\[ \alpha^3 + \beta^3 + \gamma^3 = 3\alpha\beta\gamma \quad \text{if} \quad \alpha + \beta + \gamma = 0 \]
Let:
- \( \alpha = b - c \)
- \( \beta = c - a \)
- \( \gamma = a - b \)

Notice that:
\[ \alpha + \beta + \gamma = (b - c) + (c - a) + (a - b) = 0 \]

### Step 4: Calculate the product \( \alpha \beta \gamma \)
Now we need to calculate:
\[ \alpha \beta \gamma = (b - c)(c - a)(a - b) \]

### Step 5: Substitute back into the identity
Using the identity:
\[ (b - c)^3 + (c - a)^3 + (a - b)^3 = 3(b - c)(c - a)(a - b) \]

### Step 6: Final expression
Thus, we have:
\[ \left(\frac{x}{a}\right)^3 + \left(\frac{y}{b}\right)^3 + \left(\frac{z}{c}\right)^3 = 3(b - c)(c - a)(a - b) \]

### Step 7: Express in terms of x, y, z, a, b, c
We can express this in terms of \(x\), \(y\), and \(z\):
\[ (b - c) = \frac{x}{a}, \quad (c - a) = \frac{y}{b}, \quad (a - b) = \frac{z}{c} \]
Thus:
\[ 3(b - c)(c - a)(a - b) = 3 \left(\frac{x}{a}\right) \left(\frac{y}{b}\right) \left(\frac{z}{c}\right) \]

### Conclusion
The final value is:
\[ \frac{3xyz}{abc} \]