Let `p, q, r epsilon R` be such that `2q = p + r` and `(2017-x)/(px) = (2017-y)/(qy) = (2017-z)/(rz)` , then the correct relation between x,y,z is

Options

A. `(x)/(x) = (x-y)/(y-z)`
B. `(x)/(y) = (x-y)/(y-z)`
C. `(x)/(z) = (x-y)/(y-z)`
D. `(x+z)/(y+z) = (x-z)/(y-z)`

Explanation

To solve the problem, we need to derive the relationship between \( x, y, z \) based on the given equations. Let's go through the steps systematically.

### Step 1: Understand the Given Equations
We are given two equations:
1. \( 2q = p + r \)
2. \( \frac{2017 - x}{px} = \frac{2017 - y}{qy} = \frac{2017 - z}{rz} \)

### Step 2: Set a Common Variable
Let \( k = \frac{2017 - x}{px} = \frac{2017 - y}{qy} = \frac{2017 - z}{rz} \).

From this, we can express \( x, y, z \) in terms of \( k \):
- \( 2017 - x = pkx \) → \( x + pkx = 2017 \) → \( x(1 + pk) = 2017 \) → \( x = \frac{2017}{1 + pk} \)
- \( 2017 - y = qky \) → \( y + qky = 2017 \) → \( y(1 + qk) = 2017 \) → \( y = \frac{2017}{1 + qk} \)
- \( 2017 - z = r kz \) → \( z + rkz = 2017 \) → \( z(1 + rk) = 2017 \) → \( z = \frac{2017}{1 + rk} \)

### Step 3: Substitute and Rearrange
Now we have:
- \( x = \frac{2017}{1 + pk} \)
- \( y = \frac{2017}{1 + qk} \)
- \( z = \frac{2017}{1 + rk} \)

### Step 4: Establish Relationships
From the equation \( 2q = p + r \), we can manipulate it to find relationships between \( p, q, r \):
- Rearranging gives us \( q - r = p - q \) or \( p - q = q - r \).

### Step 5: Formulate the Ratios
Now we can express the relationships:
\[ \frac{2017}{x} - 1 = pk, \quad \frac{2017}{y} - 1 = qk, \quad \frac{2017}{z} - 1 = rk \]

### Step 6: Subtract and Rearrange
Subtract the first two equations:
\[ \frac{2017}{x} - \frac{2017}{y} = pk - qk \]
This simplifies to:
\[ \frac{2017(y - x)}{xy} = k(p - q) \]

Subtract the second and third equations:
\[ \frac{2017}{y} - \frac{2017}{z} = qk - rk \]
This simplifies to:
\[ \frac{2017(z - y)}{yz} = k(q - r) \]

### Step 7: Set Up the Final Relationship
Now we can equate the two derived expressions:
\[ \frac{y - x}{xy} = \frac{z - y}{yz} \]
Cross-multiplying gives:
\[ (y - x)z = (z - y)x \]
Rearranging leads to:
\[ x - y = \frac{y - z}{z} \]

### Final Result
Thus, we arrive at the relationship:
\[ \frac{x - y}{z - y} = \frac{x}{z} \]