Let `alpha,beta` be the roots of `x^2-4x+A=0` and `gamma,delta` be the roots of `x^2-36x+B=0` .If `alpha,beta,gamma,delta` form an increasing G.P. and `A^t=B`, then value of 't' equals:

Options

A. 4
B. 5
C. 6
D. 8

Answer

To solve the problem step-by-step, we will break down the information given and use the properties of quadratic equations and geometric progressions.

### Step 1: Identify the roots from the quadratic equations
The roots of the first equation \( x^2 - 4x + A = 0 \) are \( \alpha \) and \( \beta \). According to Vieta's formulas:
- Sum of the roots: \( \alpha + \beta = 4 \)
- Product of the roots: \( \alpha \beta = A \)

The roots of the second equation \( x^2 - 36x + B = 0 \) are \( \gamma \) and \( \delta \). Again, using Vieta's formulas:
- Sum of the roots: \( \gamma + \delta = 36 \)
- Product of the roots: \( \gamma \delta = B \)

### Step 2: Express the roots in terms of a geometric progression
Since \( \alpha, \beta, \gamma, \delta \) form an increasing geometric progression (G.P.), we can express them as:
- Let \( \alpha = a \)
- Then \( \beta = ar \)
- \( \gamma = ar^2 \)
- \( \delta = ar^3 \)

where \( a \) is the first term and \( r \) is the common ratio.

### Step 3: Set up equations based on the sums of the roots
From the sum of the roots:
1. \( \alpha + \beta = a + ar = a(1 + r) = 4 \) (Equation 1)
2. \( \gamma + \delta = ar^2 + ar^3 = ar^2(1 + r) = 36 \) (Equation 2)

### Step 4: Divide Equation 2 by Equation 1
Dividing Equation 2 by Equation 1:
\[ \frac{ar^2(1 + r)}{a(1 + r)} = \frac{36}{4} \]
This simplifies to:
\[ r^2 = 9 \]
Thus, \( r = 3 \) (since \( r \) must be positive for an increasing G.P.).

### Step 5: Substitute \( r \) back to find \( a \)
Substituting \( r = 3 \) into Equation 1:
\[ a(1 + 3) = 4 \implies 4a = 4 \implies a = 1 \]

### Step 6: Find the values of \( \alpha, \beta, \gamma, \delta \)
Now substituting \( a \) and \( r \):
- \( \alpha = a = 1 \)
- \( \beta = ar = 1 \cdot 3 = 3 \)
- \( \gamma = ar^2 = 1 \cdot 3^2 = 9 \)
- \( \delta = ar^3 = 1 \cdot 3^3 = 27 \)

### Step 7: Calculate \( A \) and \( B \)
Now we can find \( A \) and \( B \):
- \( A = \alpha \beta = 1 \cdot 3 = 3 \)
- \( B = \gamma \delta = 9 \cdot 27 = 243 \)

### Step 8: Use the relationship \( A^t = B \)
We know \( A^t = B \):
\[ 3^t = 243 \]
Since \( 243 = 3^5 \), we can equate the exponents:
\[ t = 5 \]

### Final Answer
The value of \( t \) is \( 5 \).