If `a ^(2) = by + cz, b ^(2)= cz + ax, c ^(2) = ax + by, ` then the value of `(x)/(a + x) +(y)/(b + y) + (z)/(c + z)` is

Options

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

Answer

To solve the problem, we need to evaluate the expression \( \frac{x}{a+x} + \frac{y}{b+y} + \frac{z}{c+z} \) given the equations:

1. \( a^2 = by + cz \)
2. \( b^2 = cz + ax \)
3. \( c^2 = ax + by \)

### Step 1: Assume values for \( a, b, c, x, y, z \)

Let's assume \( a = b = c = 2 \) and \( x = y = z = 1 \). This is a reasonable assumption to simplify the equations.

### Step 2: Verify the equations

Now we need to verify if these values satisfy the given equations:

- **For the first equation**:
\[ a^2 = by + cz \implies 2^2 = 2 \cdot 1 + 2 \cdot 1 \implies 4 = 2 + 2 \implies 4 = 4 \quad \text{(True)} \]

- **For the second equation**:
\[ b^2 = cz + ax \implies 2^2 = 2 \cdot 1 + 2 \cdot 1 \implies 4 = 2 + 2 \implies 4 = 4 \quad \text{(True)} \]

- **For the third equation**:
\[ c^2 = ax + by \implies 2^2 = 2 \cdot 1 + 2 \cdot 1 \implies 4 = 2 + 2 \implies 4 = 4 \quad \text{(True)} \]

Since all equations are satisfied, we can proceed.

### Step 3: Substitute values into the expression

Now substitute \( x = 1 \), \( a = 2 \), \( y = 1 \), \( b = 2 \), \( z = 1 \), and \( c = 2 \) into the expression:

\[ \frac{x}{a+x} + \frac{y}{b+y} + \frac{z}{c+z} = \frac{1}{2+1} + \frac{1}{2+1} + \frac{1}{2+1} \]

### Step 4: Simplify the expression

Each term simplifies as follows:

\[ \frac{1}{2+1} = \frac{1}{3} \]

Thus, we have:

\[ \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3} = 1 \]

### Final Answer

The value of \( \frac{x}{a+x} + \frac{y}{b+y} + \frac{z}{c+z} \) is \( 1 \).

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