Let `alpha and beta` be the roots of equation `px^2 + qx + r = 0 , p != 0`.If `p,q,r` are in A.P. and `1/alpha+1/beta=4`, then the value of `|alpha-beta|` is :

Options

A. `(sqrt(34))/(9)`
B. `(2sqrt(13))/(9)`
C. `(sqrt(61))/(9)`
D. `(2sqrt(17))/(9)`

Explanation

To solve the problem step by step, we will follow the given information and derive the required value of \(|\alpha - \beta|\).

### Step 1: Identify the roots and their relationships
Given the quadratic equation \(px^2 + qx + r = 0\), the roots \(\alpha\) and \(\beta\) have the following relationships:
- \(\alpha + \beta = -\frac{q}{p}\)
- \(\alpha \beta = \frac{r}{p}\)

### Step 2: Use the given condition
We are given that:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = 4 \]
This can be rewritten using the relationships of the roots:
\[ \frac{\alpha + \beta}{\alpha \beta} = 4 \]
Substituting the expressions for \(\alpha + \beta\) and \(\alpha \beta\):
\[ \frac{-\frac{q}{p}}{\frac{r}{p}} = 4 \]
This simplifies to:
\[ -\frac{q}{r} = 4 \implies q = -4r \]

### Step 3: Use the condition of A.P.
Since \(p, q, r\) are in Arithmetic Progression (A.P.), we have:
\[ 2q = p + r \]
Substituting \(q = -4r\) into the A.P. condition:
\[ 2(-4r) = p + r \implies -8r = p + r \]
Rearranging gives:
\[ p = -9r \]

### Step 4: Substitute back to find \(\alpha + \beta\) and \(\alpha \beta\)
Now we can substitute \(p\) and \(q\) back into the expressions for \(\alpha + \beta\) and \(\alpha \beta\):
1. For \(\alpha + \beta\):
\[ \alpha + \beta = -\frac{q}{p} = -\frac{-4r}{-9r} = \frac{4}{9} \]

2. For \(\alpha \beta\):
\[ \alpha \beta = \frac{r}{p} = \frac{r}{-9r} = -\frac{1}{9} \]

### Step 5: Calculate \(|\alpha - \beta|\)
Using the formula:
\[ |\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta} \]
Substituting the values we found:
\[ |\alpha - \beta| = \sqrt{\left(\frac{4}{9}\right)^2 - 4\left(-\frac{1}{9}\right)} \]
Calculating each term:
\[ \left(\frac{4}{9}\right)^2 = \frac{16}{81} \]
\[ 4\left(-\frac{1}{9}\right) = -\frac{4}{9} = -\frac{36}{81} \]
Now substituting back:
\[ |\alpha - \beta| = \sqrt{\frac{16}{81} + \frac{36}{81}} = \sqrt{\frac{52}{81}} = \frac{\sqrt{52}}{9} = \frac{2\sqrt{13}}{9} \]

### Final Answer
Thus, the value of \(|\alpha - \beta|\) is:
\[ \boxed{\frac{2\sqrt{13}}{9}} \]