Key Concepts and Formulas

• Roots of a Quadratic Equation: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$:
  • Sum of roots: $\alpha + \beta = -\frac{b}{a}$
  • Product of roots: $\alpha \beta = \frac{c}{a}$
• Geometric Progression (G.P.): If $p, q, r$ are in G.P., then $q^2 = pr$ or equivalently, $\frac{q}{p} = \frac{r}{q}$.
• Algebraic Identity: $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$

Step-by-Step Solution

Step 1: Expressing the Root Relationship in terms of Coefficients

We are given that $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}$. We want to rewrite this in terms of the coefficients of the quadratic. Combining the fractions, we have: $\frac{\alpha + \beta}{\alpha \beta} = \frac{3}{4}$ Substituting the expressions for the sum and product of the roots, $\alpha + \beta = -\frac{q}{p}$ and $\alpha \beta = -\frac{r}{p}$, we get: $\frac{-\frac{q}{p}}{-\frac{r}{p}} = \frac{3}{4}$ Simplifying, we have: $\frac{q}{r} = \frac{3}{4} \quad (*)$ This equation relates the coefficients $q$ and $r$.

Step 2: Incorporating the Geometric Progression Property

Since $p, q, r$ are in G.P., we know that $q^2 = pr$. Also, we can write $q = pk$ and $r = qk = pk^2$, where $k$ is the common ratio. We substitute $q = pk$ and $r = pk^2$ into equation $(*)$: $\frac{pk}{pk^2} = \frac{3}{4}$ Simplifying, we have: $\frac{1}{k} = \frac{3}{4}$ Solving for $k$, we get: $k = \frac{4}{3}$ Since the G.P. is non-constant, $k \neq 1$, which is consistent with our result.

Step 3: Calculating $(\alpha - \beta)^2$

We want to find $(\alpha - \beta)^2$. Using the identity $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$, we can write: $(\alpha - \beta)^2 = \left(-\frac{q}{p}\right)^2 - 4\left(-\frac{r}{p}\right)$ $(\alpha - \beta)^2 = \frac{q^2}{p^2} + \frac{4r}{p}$ Since $p, q, r$ are in G.P., we have $q^2 = pr$. Substituting this into the expression above: $(\alpha - \beta)^2 = \frac{pr}{p^2} + \frac{4r}{p} = \frac{r}{p} + \frac{4r}{p} = \frac{5r}{p}$ Now, we know $r = qk$ and $q = pk$, so $r = pk^2$. Substituting this into the expression: $(\alpha - \beta)^2 = \frac{5pk^2}{p} = 5k^2$ We found that $k = \frac{4}{3}$, so: $(\alpha - \beta)^2 = 5\left(\frac{4}{3}\right)^2 = 5\left(\frac{16}{9}\right) = \frac{80}{9}$

Common Mistakes & Tips

• Sign Errors: Be careful with the signs when using the sum and product of roots formulas, especially with the constant term being $-r$.
• G.P. vs. A.P.: Ensure you use the correct properties for a Geometric Progression ($q^2 = pr$).
• Algebraic Manipulation: Double-check your algebraic steps, especially when squaring terms or combining fractions.

Summary

We used the relationships between the roots and coefficients of a quadratic equation, along with the properties of a geometric progression, to find the value of $(\alpha - \beta)^2$. We first expressed the given condition $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}$ in terms of the coefficients. Then, using the G.P. property, we found the common ratio $k$. Finally, we substituted these values into the expression for $(\alpha - \beta)^2$ to obtain the answer.

The final answer is $\boxed{\frac{80}{9}}$, which corresponds to option (D).