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

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the given information
We know that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( px^2 + qx + r = 0 \). We are given that \( p, q, r \) are in Arithmetic Progression (A.P.) and that \( \frac{1}{\alpha} + \frac{1}{\beta} = 4 \).

### Step 2: Relate the roots to the coefficients
From Vieta's formulas, we know:
- \( \alpha + \beta = -\frac{q}{p} \)
- \( \alpha \beta = \frac{r}{p} \)

### Step 3: Use the given condition \( \frac{1}{\alpha} + \frac{1}{\beta} = 4 \)
We can rewrite this as:
\[ \frac{\alpha + \beta}{\alpha \beta} = 4 \]
Substituting the expressions from Vieta's formulas:
\[ \frac{-\frac{q}{p}}{\frac{r}{p}} = 4 \]
This simplifies to:
\[ -\frac{q}{r} = 4 \quad \Rightarrow \quad q = -4r \]

### Step 4: Use the condition that \( p, q, r \) are in A.P.
Since \( p, q, r \) are in A.P., we have:
\[ 2q = p + r \]
Substituting \( q = -4r \):
\[ 2(-4r) = p + r \quad \Rightarrow \quad -8r = p + r \quad \Rightarrow \quad p = -9r \]

### Step 5: Substitute \( p \) and \( q \) back into the equations
Now we have:
- \( p = -9r \)
- \( q = -4r \)

### Step 6: Find \( \alpha + \beta \) and \( \alpha \beta \)
Using the values of \( p \) and \( q \):
\[ \alpha + \beta = -\frac{q}{p} = -\frac{-4r}{-9r} = \frac{4}{9} \]
\[ \alpha \beta = \frac{r}{p} = \frac{r}{-9r} = -\frac{1}{9} \]

### Step 7: Calculate \( |\alpha - \beta| \)
We know:
\[ |\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta} \]
Substituting the values we found:
\[ |\alpha - \beta| = \sqrt{\left(\frac{4}{9}\right)^2 - 4\left(-\frac{1}{9}\right)} \]
Calculating:
\[ = \sqrt{\frac{16}{81} + \frac{4}{9}} = \sqrt{\frac{16}{81} + \frac{36}{81}} = \sqrt{\frac{52}{81}} = \frac{\sqrt{52}}{9} = \frac{2\sqrt{13}}{9} \]

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