`x_(1) and x_(2)` are two variates with variances `sigma_(1)^(2) and sigma_(2)^(2)` respectively and r is the correlation between them. The value of a such that `x_(1) + ax_(2)` and `x_(1)+(sigma_(1))/(sigma_(2))x_(2)` are uncorrelated is

Options

A. `-(sigma_(1))/(sigma_(2))`
B. `r(sigma_(2))/(sigma_(1))`
C. `sigma_(1)sigma_(2)`
D. none of these

Answer

To solve the problem, we need to find the value of \( a \) such that the variates \( x_1 + ax_2 \) and \( x_1 + \frac{\sigma_1}{\sigma_2} x_2 \) are uncorrelated.

### Step-by-Step Solution:

1. **Understanding Uncorrelated Variables**:
Two random variables are uncorrelated if their covariance is zero. Therefore, we need to find the covariance of \( x_1 + ax_2 \) and \( x_1 + \frac{\sigma_1}{\sigma_2} x_2 \) and set it to zero.

2. **Using the Covariance Formula**:
The covariance of two linear combinations of random variables can be expressed as:
\[ \text{Cov}(A, B) = \text{Cov}(x_1 + ax_2, x_1 + \frac{\sigma_1}{\sigma_2} x_2) \]
Expanding this using the properties of covariance:
\[ \text{Cov}(x_1 + ax_2, x_1 + \frac{\sigma_1}{\sigma_2} x_2) = \text{Cov}(x_1, x_1) + \text{Cov}(x_1, \frac{\sigma_1}{\sigma_2} x_2) + \text{Cov}(ax_2, x_1) + \text{Cov}(ax_2, \frac{\sigma_1}{\sigma_2} x_2) \]

3. **Substituting Covariance Values**:
Using the fact that \( \text{Cov}(x_1, x_1) = \text{Var}(x_1) = \sigma_1^2 \), and \( \text{Cov}(x_1, x_2) = r \sigma_1 \sigma_2 \):
\[ = \sigma_1^2 + \frac{\sigma_1}{\sigma_2} \text{Cov}(x_1, x_2) + a \text{Cov}(x_2, x_1) + a \frac{\sigma_1}{\sigma_2} \text{Cov}(x_2, x_2) \]
\[ = \sigma_1^2 + \frac{\sigma_1}{\sigma_2} (r \sigma_1 \sigma_2) + a (r \sigma_1 \sigma_2) + a \frac{\sigma_1}{\sigma_2} \sigma_2^2 \]
\[ = \sigma_1^2 + r \sigma_1^2 + a r \sigma_1 \sigma_2 + a \sigma_1 \sigma_2 \]

4. **Setting Covariance to Zero**:
Now, we set the above expression to zero:
\[ \sigma_1^2 (1 + r) + a \sigma_1 \sigma_2 (r + 1) = 0 \]

5. **Solving for \( a \)**:
Rearranging gives:
\[ a \sigma_1 \sigma_2 (r + 1) = -\sigma_1^2 (1 + r) \]
Dividing both sides by \( \sigma_1 \sigma_2 (r + 1) \) (assuming \( r \neq -1 \)):
\[ a = -\frac{\sigma_1}{\sigma_2} \]

### Final Answer:
The value of \( a \) such that \( x_1 + ax_2 \) and \( x_1 + \frac{\sigma_1}{\sigma_2} x_2 \) are uncorrelated is:
\[ \boxed{-\frac{\sigma_1}{\sigma_2}} \]