If the imaginary part of `(2 + i)/( ai - 1)` is zero where ` a in R ` then a =

Options

A. `1//2`
B. 2
C. `-1//2`
D. `-2`

Answer

To solve the problem, we need to find the value of \( a \) such that the imaginary part of the complex number \( \frac{2 + i}{ai - 1} \) is zero. Here are the steps to find the solution:

### Step 1: Write the expression
We start with the expression:
\[ z = \frac{2 + i}{ai - 1} \]

### Step 2: Multiply by the conjugate
To eliminate the imaginary part from the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( ai + 1 \):
\[ z = \frac{(2 + i)(ai + 1)}{(ai - 1)(ai + 1)} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ (ai - 1)(ai + 1) = a^2i^2 - 1 = -a^2 - 1 \]

### Step 4: Expand the numerator
Now, we expand the numerator:
\[ (2 + i)(ai + 1) = 2ai + 2 + ai^2 + i = 2 + 2ai - a \]
Since \( i^2 = -1 \), we have:
\[ ai^2 = -a \Rightarrow 2 + 2ai - a \]

### Step 5: Combine terms
Thus, we can write the expression for \( z \) as:
\[ z = \frac{(2 - a) + (2a + 1)i}{-a^2 - 1} \]

### Step 6: Identify the imaginary part
The imaginary part of \( z \) is given by:
\[ \text{Imaginary part} = \frac{2a + 1}{-a^2 - 1} \]

### Step 7: Set the imaginary part to zero
To find \( a \), we set the imaginary part equal to zero:
\[ \frac{2a + 1}{-a^2 - 1} = 0 \]

### Step 8: Solve for \( a \)
The fraction is zero when the numerator is zero:
\[ 2a + 1 = 0 \]
Solving for \( a \):
\[ 2a = -1 \Rightarrow a = -\frac{1}{2} \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{-\frac{1}{2}} \]