Question

If \(\alpha,\beta\) are the roots of \(x^2-3x+a=0\), and \(\gamma,\delta\) are the roots of \(x^2-12x+b=0\), and the numbers \(\alpha,\beta,\gamma,\delta\) in order form an increasing G.P., then:

• A. \(a=3,b=12\)
• B. \(a=12,b=3\)
• C. \(a=2,b=32\)
• D. \(a=4,b=16\)

Hint:

If four numbers are in G.P., write them as \(k,kr,kr^2,kr^3\). Then use sum and product of roots to find the unknown constants.

Answer 1

The Correct Option is C

Concept:
This question is based on two concepts: \[ \text{Sum of roots and product of roots of a quadratic equation} \] and \[ \text{Geometric Progression} \] For a quadratic equation: \[ x^2-Sx+P=0 \] if its roots are \(r_1\) and \(r_2\), then: \[ r_1+r_2=S \] and \[ r_1r_2=P \] Also, if four numbers are in G.P., then they can be written as: \[ k,\ kr,\ kr^2,\ kr^3 \] where \(r\) is the common ratio.

Step 1: Use the first quadratic equation.
Given: \[ x^2-3x+a=0 \] Its roots are: \[ \alpha,\beta \] Therefore, by sum of roots: \[ \alpha+\beta=3 \] and by product of roots: \[ \alpha\beta=a \]

Step 2: Use the second quadratic equation.
Given: \[ x^2-12x+b=0 \] Its roots are: \[ \gamma,\delta \] Therefore: \[ \gamma+\delta=12 \] and: \[ \gamma\delta=b \]

Step 3: Write \(\alpha,\beta,\gamma,\delta\) as an increasing G.P.
Since \(\alpha,\beta,\gamma,\delta\) in order form an increasing G.P., let: \[ \alpha=k \] \[ \beta=kr \] \[ \gamma=kr^2 \] \[ \delta=kr^3 \] where \(r>1\), because the G.P. is increasing.

Step 4: Use the sum of first two roots.
Since: \[ \alpha+\beta=3 \] we get: \[ k+kr=3 \] \[ k(1+r)=3 \]

Step 5: Use the sum of next two roots.
Since: \[ \gamma+\delta=12 \] we get: \[ kr^2+kr^3=12 \] Taking \(kr^2\) common: \[ kr^2(1+r)=12 \]

Step 6: Divide the two equations.
We have: \[ k(1+r)=3 \] and: \[ kr^2(1+r)=12 \] Now divide the second equation by the first equation: \[ \frac{kr^2(1+r)}{k(1+r)}=\frac{12}{3} \] After cancellation: \[ r^2=4 \] \[ r=2 \] Since the G.P. is increasing: \[ r=2 \]

Step 7: Find the value of \(k\).
Using: \[ k(1+r)=3 \] Substitute: \[ r=2 \] \[ k(1+2)=3 \] \[ 3k=3 \] \[ k=1 \]

Step 8: Find the four roots.
The G.P. terms are: \[ k,\ kr,\ kr^2,\ kr^3 \] Substitute \(k=1\) and \(r=2\): \[ \alpha=1 \] \[ \beta=2 \] \[ \gamma=4 \] \[ \delta=8 \] So: \[ \alpha,\beta,\gamma,\delta=1,2,4,8 \]

Step 9: Find \(a\).
From the first quadratic equation: \[ a=\alpha\beta \] \[ a=1\times 2 \] \[ a=2 \]

Step 10: Find \(b\).
From the second quadratic equation: \[ b=\gamma\delta \] \[ b=4\times 8 \] \[ b=32 \] Hence, the correct answer is: \[ \boxed{(C)\ a=2,b=32} \]