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

Options

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

Answer

To solve the problem, we need to find the value of 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: Rewrite the expression

We can rewrite the expression as:

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

This gives us:

\[ \frac{ax}{a^2 + ax} + \frac{by}{b^2 + by} + \frac{cz}{c^2 + cz} \]

### Step 2: Substitute the values of \(a^2\), \(b^2\), and \(c^2\)

Using the equations given, we can substitute \(a^2\), \(b^2\), and \(c^2\):

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

### Step 3: Substitute into the expression

Substituting these into our rewritten expression gives us:

\[ \frac{ax}{by + cz + ax} + \frac{by}{cz + ax + by} + \frac{cz}{ax + by + cz} \]

### Step 4: Simplify the expression

Notice that each term in the denominator can be rewritten as:

- For the first term: \(by + cz + ax = a^2 + ax\)
- For the second term: \(cz + ax + by = b^2 + by\)
- For the third term: \(ax + by + cz = c^2 + cz\)

Thus, we can rewrite the expression as:

\[ \frac{ax}{a^2 + ax} + \frac{by}{b^2 + by} + \frac{cz}{c^2 + cz} \]

### Step 5: Recognize the pattern

Notice that each fraction simplifies to:

\[ \frac{ax}{a^2 + ax} = \frac{ax}{a(a+x)} = \frac{x}{a+x} \]

This means that each term simplifies similarly, leading to:

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

### Step 6: Conclusion

Since we have shown that the expression simplifies back to itself, we can conclude that the value of the original expression is:

\[ 1 \]