`alpha ,beta` be the roots of the equation `x^2 – 3x + a =0` and `gamma , delta` the roots of `x^2 – 12x + b =0` and numbers `alpha,beta,gamma,delta` (in this order) form an increasing G.P., then

Options

A. `a=3,b=12`
B. `a=12,b=3`
C. `a=2,b=32`
D. `a=4,b=16`.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the roots \( \alpha, \beta, \gamma, \delta \) form an increasing geometric progression (G.P.).

### Step 1: Identify the roots of the equations
The roots of the equation \( x^2 - 3x + a = 0 \) can be found using the quadratic formula:
\[ \alpha, \beta = \frac{3 \pm \sqrt{3^2 - 4a}}{2} \]
This simplifies to:
\[ \alpha, \beta = \frac{3 \pm \sqrt{9 - 4a}}{2} \]

### Step 2: Find the roots of the second equation
Similarly, the roots of the equation \( x^2 - 12x + b = 0 \) are:
\[ \gamma, \delta = \frac{12 \pm \sqrt{12^2 - 4b}}{2} \]
This simplifies to:
\[ \gamma, \delta = \frac{12 \pm \sqrt{144 - 4b}}{2} \]

### Step 3: Set up the G.P. condition
Since \( \alpha, \beta, \gamma, \delta \) form a G.P., we can express them in terms of \( \alpha \):
Let \( \alpha = A \), \( \beta = Ar \), \( \gamma = Ar^2 \), and \( \delta = Ar^3 \) for some common ratio \( r \).

### Step 4: Use the relationships from the roots
From the properties of roots:
1. For \( \alpha + \beta = 3 \):
\[ A + Ar = 3 \implies A(1 + r) = 3 \tag{1} \]
2. For \( \gamma + \delta = 12 \):
\[ Ar^2 + Ar^3 = 12 \implies Ar^2(1 + r) = 12 \tag{2} \]

### Step 5: Solve for \( A \) and \( r \)
From equation (1):
\[ A = \frac{3}{1 + r} \]
Substituting this into equation (2):
\[ \frac{3r^2}{1 + r}(1 + r) = 12 \implies 3r^2 = 12 \implies r^2 = 4 \implies r = 2 \text{ (since it's a G.P., we take the positive root)} \]

### Step 6: Find \( A \)
Substituting \( r = 2 \) back into equation (1):
\[ A(1 + 2) = 3 \implies 3A = 3 \implies A = 1 \]

### Step 7: Find the roots
Now we can find the roots:
\[ \alpha = A = 1, \quad \beta = Ar = 1 \cdot 2 = 2, \quad \gamma = Ar^2 = 1 \cdot 4 = 4, \quad \delta = Ar^3 = 1 \cdot 8 = 8 \]

### Step 8: Find values of \( a \) and \( b \)
Using the roots:
1. For \( \alpha \) and \( \beta \):
\[ \alpha + \beta = 3 \implies 1 + 2 = 3 \text{ (correct)} \]
\[ \alpha \beta = a \implies 1 \cdot 2 = 2 \implies a = 2 \]
2. For \( \gamma \) and \( \delta \):
\[ \gamma + \delta = 12 \implies 4 + 8 = 12 \text{ (correct)} \]
\[ \gamma \delta = b \implies 4 \cdot 8 = 32 \implies b = 32 \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ a = 2, \quad b = 32 \]