Question

If \( \alpha, \beta \) are the roots of the equation \( x^2 - 4x + p = 0 \) and \( \gamma, \delta \) are the roots of the equation \( x^2 - x + q = 0 \). When \( \alpha, \beta, \gamma, \delta \) form a G.P. with positive common ratio, then the value of \( (p + q) \) equals:

• A. \( \frac{22}{9} \)
• B. \( \frac{33}{9} \)
• C. \( \frac{21}{9} \)
• D. \( \frac{34}{9} \)

Hint:

When roots of multiple equations are in A.P. or G.P., express them using a single variable and the common difference/ratio. Dividing the sum equations is the fastest way to find the ratio $r$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Let the roots in G.P. be \( a, ar, ar^2, ar^3 \). We relate these to the coefficients of the quadratic equations using sum and product of roots.
Step 2: Key Formula or Approach:
1. \( \alpha + \beta = 4, \alpha \beta = p \)
2. \( \gamma + \delta = 1, \gamma \delta = q \)
Step 3: Detailed Explanation:
1. Equations for sums: \[ a + ar = 4 \implies a(1+r) = 4 \] \[ ar^2 + ar^3 = 1 \implies ar^2(1+r) = 1 \] 2. Divide the equations: \[ \frac{ar^2(1+r)}{a(1+r)} = \frac{1}{4} \implies r^2 = \frac{1}{4} \implies r = \frac{1}{2} \text{ (since } r > 0) \] 3. Find \( a \): \[ a\left(1 + \frac{1}{2}\right) = 4 \implies a\left(\frac{3}{2}\right) = 4 \implies a = \frac{8}{3} \] 4. Roots are: \( \alpha = \frac{8}{3}, \beta = \frac{4}{3}, \gamma = \frac{2}{3}, \delta = \frac{1}{3} \). 5. Calculate \( p \) and \( q \): \[ p = \alpha \beta = \frac{8}{3} \cdot \frac{4}{3} = \frac{32}{9} \] \[ q = \gamma \delta = \frac{2}{3} \cdot \frac{1}{3} = \frac{2}{9} \] 6. \( p + q = \frac{32}{9} + \frac{2}{9} = \frac{34}{9} \).
Step 4: Final Answer:
The value of \( (p+q) \) is \( \frac{34}{9} \).