If `(p)/( q - r) = (p + q)/( r) = (q)/( p)`, Then find `q : p : r`

Options

A. `1 : 2 : 3 `
B. `3 : 2 : 1`
C. ` 4 : 2 : 3 `
D. ` 2 : 4 : 7`

Answer

To solve the equation \( \frac{p}{q - r} = \frac{p + q}{r} = \frac{q}{p} \), we will denote the common ratio as \( k \). Therefore, we can write the following equations:

1. \( \frac{p}{q - r} = k \)
2. \( \frac{p + q}{r} = k \)
3. \( \frac{q}{p} = k \)

### Step 1: Express \( p \), \( q \), and \( r \) in terms of \( k \)

From the first equation:
\[ p = k(q - r) \quad \text{(1)} \]

From the second equation:
\[ p + q = kr \quad \text{(2)} \]

From the third equation:
\[ q = kp \quad \text{(3)} \]

### Step 2: Substitute \( q \) from equation (3) into equations (1) and (2)

Substituting \( q = kp \) into equation (1):
\[ p = k(kp - r) \]
This simplifies to:
\[ p = k^2p - kr \]
Rearranging gives:
\[ p - k^2p = -kr \]
\[ p(1 - k^2) = -kr \]
Thus,
\[ r = -\frac{p(1 - k^2)}{k} \quad \text{(4)} \]

Now substitute \( q = kp \) into equation (2):
\[ p + kp = kr \]
This simplifies to:
\[ p(1 + k) = kr \]
Substituting \( r \) from equation (4):
\[ p(1 + k) = k\left(-\frac{p(1 - k^2)}{k}\right) \]
This simplifies to:
\[ p(1 + k) = -p(1 - k^2) \]
Assuming \( p \neq 0 \), we can divide both sides by \( p \):
\[ 1 + k = -(1 - k^2) \]
This simplifies to:
\[ 1 + k = -1 + k^2 \]
Rearranging gives:
\[ k^2 - k - 2 = 0 \]

### Step 3: Solve the quadratic equation

Factoring the quadratic:
\[ (k - 2)(k + 1) = 0 \]
Thus, \( k = 2 \) or \( k = -1 \).

### Step 4: Find \( q \), \( p \), and \( r \) for \( k = 2 \)

Using \( k = 2 \):
From equation (3):
\[ q = 2p \]

Substituting \( k = 2 \) into equation (4):
\[ r = -\frac{p(1 - 2^2)}{2} = -\frac{p(-3)}{2} = \frac{3p}{2} \]

### Step 5: Find the ratio \( q : p : r \)

Now we have:
- \( q = 2p \)
- \( p = p \)
- \( r = \frac{3p}{2} \)

Thus, the ratio \( q : p : r \) is:
\[ 2p : p : \frac{3p}{2} \]
Dividing by \( p \):
\[ 2 : 1 : \frac{3}{2} \]
To eliminate the fraction, multiply the entire ratio by 2:
\[ 4 : 2 : 3 \]

### Final Answer
The required ratio \( q : p : r \) is \( 4 : 2 : 3 \).

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