If the coefficients of `x^2 and x^3` in the expansion of `(3 + ax)^(9)` be same, then the value of `a` is

Options

A. (a) `(3)/(7)`
B. (b) `(7)/(3)`
C. (c) `(7)/(9)`
D. (d) `(9)/(7)`

Answer

To find the value of \( a \) such that the coefficients of \( x^2 \) and \( x^3 \) in the expansion of \( (3 + ax)^9 \) are equal, we can follow these steps:

### Step 1: Identify the General Term
The general term in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 3 \), \( b = ax \), and \( n = 9 \). Thus, the general term becomes:
\[ T_{r+1} = \binom{9}{r} (3)^{9-r} (ax)^r = \binom{9}{r} 3^{9-r} a^r x^r \]

### Step 2: Find the Coefficient of \( x^2 \)
To find the coefficient of \( x^2 \), we set \( r = 2 \):
\[ T_{3} = \binom{9}{2} 3^{9-2} a^2 x^2 \]
The coefficient of \( x^2 \) is:
\[ \text{Coefficient of } x^2 = \binom{9}{2} 3^7 a^2 \]

### Step 3: Find the Coefficient of \( x^3 \)
To find the coefficient of \( x^3 \), we set \( r = 3 \):
\[ T_{4} = \binom{9}{3} 3^{9-3} a^3 x^3 \]
The coefficient of \( x^3 \) is:
\[ \text{Coefficient of } x^3 = \binom{9}{3} 3^6 a^3 \]

### Step 4: Set the Coefficients Equal
According to the problem, the coefficients of \( x^2 \) and \( x^3 \) are equal:
\[ \binom{9}{2} 3^7 a^2 = \binom{9}{3} 3^6 a^3 \]

### Step 5: Simplify the Equation
We can simplify this equation by dividing both sides by \( 3^6 \):
\[ \binom{9}{2} 3 a^2 = \binom{9}{3} a^3 \]
Next, we can express \( \binom{9}{2} \) and \( \binom{9}{3} \):
\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]
Substituting these values in:
\[ 36 \times 3 a^2 = 84 a^3 \]
This simplifies to:
\[ 108 a^2 = 84 a^3 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 84 a^3 - 108 a^2 = 0 \]
Factoring out \( a^2 \):
\[ a^2 (84 a - 108) = 0 \]
This gives us two cases:
1. \( a^2 = 0 \) (which implies \( a = 0 \), but we discard this as it does not satisfy the original problem)
2. \( 84 a - 108 = 0 \)

### Step 7: Solve for \( a \)
Solving \( 84 a - 108 = 0 \):
\[ 84 a = 108 \implies a = \frac{108}{84} = \frac{9}{7} \]

### Conclusion
Thus, the value of \( a \) is:
\[ \boxed{\frac{9}{7}} \]