Key Concepts and Formulas

• Arithmetic Progression (A.P.): If $a, b, c$ are in A.P., then $2b = a + c$.
• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, $\alpha + \beta = -\frac{b}{a}$ and $\alpha \beta = \frac{c}{a}$.
• Difference of Roots: For a quadratic equation $ax^2 + bx + c = 0$, $|\alpha - \beta| = \frac{\sqrt{b^2 - 4ac}}{|a|}$.

Step-by-Step Solution

Step 1: Apply the Arithmetic Progression condition.

We are given that $p, q, r$ are in A.P. Therefore, using the property of A.P., we have: $2q = p + r \quad \cdots (1)$ Explanation: This equation establishes the relationship between the coefficients based on the given arithmetic progression.

Step 2: Simplify the given relationship between reciprocals of the roots.

We are given that $\frac{1}{\alpha} + \frac{1}{\beta} = 4$. Combining the fractions, we get: $\frac{\alpha + \beta}{\alpha \beta} = 4$ Explanation: This step rewrites the given condition in terms of the sum and product of the roots, preparing it for the application of Vieta's formulas.

Step 3: Apply Vieta's Formulas to relate the coefficients and roots.

For the quadratic equation $px^2 + qx + r = 0$, Vieta's formulas give us: $\alpha + \beta = -\frac{q}{p}$ $\alpha \beta = \frac{r}{p}$ Substituting these into the equation from Step 2, we get: $\frac{-\frac{q}{p}}{\frac{r}{p}} = 4$ Simplifying, we have: $-\frac{q}{r} = 4$ Thus, $q = -4r \quad \cdots (2)$ Explanation: Here, we use Vieta's formulas to connect the sum and product of the roots to the coefficients of the quadratic equation, resulting in a relationship between $q$ and $r$.

Step 4: Find the relationship between $p$ and $r$.

Substitute equation (2) into equation (1): $2(-4r) = p + r$ $-8r = p + r$ $p = -9r \quad \cdots (3)$ Explanation: Substituting the value of q in terms of r into the equation obtained from the arithmetic progression allows us to express p in terms of r as well.

Step 5: Calculate the absolute difference between the roots, $|\alpha - \beta|$.

Using the formula for the difference of roots and substituting $q = -4r$ and $p = -9r$, we have: $|\alpha - \beta| = \frac{\sqrt{q^2 - 4pr}}{|p|} = \frac{\sqrt{(-4r)^2 - 4(-9r)(r)}}{|-9r|}$ $|\alpha - \beta| = \frac{\sqrt{16r^2 + 36r^2}}{9|r|} = \frac{\sqrt{52r^2}}{9|r|}$ $|\alpha - \beta| = \frac{\sqrt{4 \cdot 13 \cdot r^2}}{9|r|} = \frac{2\sqrt{13}|r|}{9|r|}$ Since $r \ne 0$ (because $p \ne 0$), we can cancel $|r|$: $|\alpha - \beta| = \frac{2\sqrt{13}}{9}$ Explanation: Substituting the values of p and q in terms of r into the formula for the difference of roots allows us to calculate its value. Simplifying the expression, we get a constant value, independent of r.

Common Mistakes & Tips

• Ensure correct application of Vieta's formulas, paying attention to signs.
• Remember to simplify the radical expression completely.
• Avoid sign errors during substitution and simplification.

Summary

By using the properties of arithmetic progressions and Vieta's formulas, we established relationships between the coefficients of the quadratic equation. Substituting these relationships into the formula for the difference of the roots, we found the value of $|\alpha - \beta|$.

The final answer is $\boxed{\frac{2\sqrt{13}}{9}}$, which corresponds to option (B).