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 terms in Geometric Progression (GP) be \(\alpha = a, \beta = ar, \gamma = ar^2, \delta = ar^3\), where \(r>0\).
We use Vieta's formulas for the sum and product of roots for both quadratic equations.
Step 2: Key Formula or Approach:
For \(x^2 - 4x + p = 0\):
Sum of roots: \(\alpha + \beta = 4 \Rightarrow a + ar = 4 \Rightarrow a(1 + r) = 4\) ---(1)
Product of roots: \(\alpha\beta = p \Rightarrow a^2r = p\) ---(2)
For \(x^2 - x + q = 0\):
Sum of roots: \(\gamma + \delta = 1 \Rightarrow ar^2 + ar^3 = 1 \Rightarrow ar^2(1 + r) = 1\) ---(3)
Product of roots: \(\gamma\delta = q \Rightarrow a^2r^5 = q\) ---(4)
Step 3: Detailed Explanation:
Divide equation (3) by equation (1):
\[ \frac{ar^2(1 + r)}{a(1 + r)} = \frac{1}{4} \Rightarrow r^2 = \frac{1}{4} \]
Since common ratio \(r\) is positive: \(r = \frac{1}{2}\).
Substitute \(r = 1/2\) into equation (1):
\[ a\left(1 + \frac{1}{2}\right) = 4 \Rightarrow a \cdot \frac{3}{2} = 4 \Rightarrow a = \frac{8}{3} \]
Now, calculate \(p\) from equation (2):
\[ p = a^2r = \left(\frac{8}{3}\right)^2 \cdot \frac{1}{2} = \frac{64}{9} \cdot \frac{1}{2} = \frac{32}{9} \]
Calculate \(q\) from equation (4):
\[ q = a^2r^5 = \left(\frac{8}{3}\right)^2 \cdot \left(\frac{1}{2}\right)^5 = \frac{64}{9} \cdot \frac{1}{32} = \frac{2}{9} \]
Finally, find \((p + q)\):
\[ p + q = \frac{32}{9} + \frac{2}{9} = \frac{34}{9} \]
Step 4: Final Answer:
The value of \(p + q\) is \(\frac{34}{9}\).