If ` (x)/( xa + yb + zx) = (y)/( y a + zb + xc) = (z)/( za + xb + yc) and x + y + z ne 0` then each ratio is

Options

A. ` (1)/( a - b - c) `
B. `(1)/( a + b - c)`
C. `(1)/( a - b + c)`
D. `(1)/( a + b + c)`

Answer

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

\[ \frac{x}{xa + yb + zc} = \frac{y}{ya + zb + xc} = \frac{z}{za + xb + yc} \]

Let us denote the common ratio by \( k \). Therefore, we can write:

1. \( \frac{x}{xa + yb + zc} = k \)
2. \( \frac{y}{ya + zb + xc} = k \)
3. \( \frac{z}{za + xb + yc} = k \)

From these equations, we can express \( x, y, z \) in terms of \( k \):

### Step 1: Express \( x \)
From the first equation:
\[ x = k(xa + yb + zc) \]
This can be rewritten as:
\[ x = kxa + kyb + kzc \]
Rearranging gives:
\[ x - kxa = kyb + kzc \]
Factoring out \( x \):
\[ x(1 - ka) = kyb + kzc \]
Thus, we have:
\[ x = \frac{kyb + kzc}{1 - ka} \quad \text{(1)} \]

### Step 2: Express \( y \)
From the second equation:
\[ y = k(ya + zb + xc) \]
This can be rewritten as:
\[ y = kya + kzb + kxc \]
Rearranging gives:
\[ y - kya = kzb + kxc \]
Factoring out \( y \):
\[ y(1 - ka) = kzb + kxc \]
Thus, we have:
\[ y = \frac{kzb + kxc}{1 - kb} \quad \text{(2)} \]

### Step 3: Express \( z \)
From the third equation:
\[ z = k(za + xb + yc) \]
This can be rewritten as:
\[ z = kza + kxb + kyc \]
Rearranging gives:
\[ z - kza = kxb + kyc \]
Factoring out \( z \):
\[ z(1 - ka) = kxb + kyc \]
Thus, we have:
\[ z = \frac{kxb + kyc}{1 - kc} \quad \text{(3)} \]

### Step 4: Establish the ratios
Now we can see that we have expressed \( x, y, z \) in terms of \( k \). Since \( x + y + z \neq 0 \), we can analyze the ratios:

From equations (1), (2), and (3), we can conclude that:

\[ \frac{x}{y} = \frac{yb + zc}{zb + xc} \]
\[ \frac{y}{z} = \frac{zb + xc}{xc + ya} \]
\[ \frac{z}{x} = \frac{xc + ya}{yb + zc} \]

### Final Conclusion
Since all three ratios are equal to \( k \), we can conclude that:

\[ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} = k \]

Thus, each ratio is equal to a constant \( k \).