In expansion of `(x+a)^(5), T_(2):T_(3)=1:3`, then `x:a` is equal to

Options

A. 1 : 2
B. 2 : 1
C. 2 : 3
D. 3 : 2

Answer

To solve the problem, we need to find the ratio \( \frac{x}{a} \) given that the ratio of the second term \( T_2 \) to the third term \( T_3 \) in the expansion of \( (x + a)^5 \) is \( 1:3 \).

### Step-by-Step Solution:

1. **Identify the General Term**:
The \( r \)-th term in the expansion of \( (x + a)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} x^{n-r} a^r \]
Here, \( n = 5 \).

2. **Calculate \( T_2 \)**:
The second term \( T_2 \) corresponds to \( r = 1 \):
\[ T_2 = \binom{5}{1} x^{5-1} a^1 = 5 x^4 a \]

3. **Calculate \( T_3 \)**:
The third term \( T_3 \) corresponds to \( r = 2 \):
\[ T_3 = \binom{5}{2} x^{5-2} a^2 = 10 x^3 a^2 \]

4. **Set Up the Ratio**:
According to the problem, the ratio \( \frac{T_2}{T_3} = \frac{1}{3} \):
\[ \frac{5 x^4 a}{10 x^3 a^2} = \frac{1}{3} \]

5. **Simplify the Ratio**:
Simplifying the left-hand side:
\[ \frac{5 x^4 a}{10 x^3 a^2} = \frac{5}{10} \cdot \frac{x^4}{x^3} \cdot \frac{1}{a} = \frac{1}{2} \cdot \frac{x}{a} \]

6. **Set Up the Equation**:
Now we have:
\[ \frac{1}{2} \cdot \frac{x}{a} = \frac{1}{3} \]

7. **Cross Multiply to Solve for \( \frac{x}{a} \)**:
Cross multiplying gives:
\[ 3 \cdot \frac{x}{a} = 2 \implies \frac{x}{a} = \frac{2}{3} \]

### Final Result:
Thus, the ratio \( \frac{x}{a} \) is equal to \( \frac{2}{3} \).