To evaluate the expression \(\frac{i}{2-i}\), we will follow these steps:
### Step 1: Multiply by the Conjugate
To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 - i\) is \(2 + i\).
\[
\frac{i}{2 - i} \cdot \frac{2 + i}{2 + i} = \frac{i(2 + i)}{(2 - i)(2 + i)}
\]
### Step 2: Simplify the Numerator
Now, we will simplify the numerator:
\[
i(2 + i) = 2i + i^2
\]
Since \(i^2 = -1\), we can substitute this in:
\[
2i + i^2 = 2i - 1
\]
### Step 3: Simplify the Denominator
Next, we simplify the denominator using the difference of squares:
\[
(2 - i)(2 + i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5
\]
### Step 4: Combine the Results
Now we can combine the results from the numerator and the denominator:
\[
\frac{2i - 1}{5}
\]
### Step 5: Separate Real and Imaginary Parts
We can separate the real and imaginary parts:
\[
\frac{2i - 1}{5} = \frac{-1}{5} + \frac{2}{5}i
\]
### Final Answer
Thus, the final result is:
\[
\frac{i}{2 - i} = -\frac{1}{5} + \frac{2}{5}i
\]
### Conclusion
The correct option is \(c: -\frac{1}{5} + \frac{2}{5}i\).
---
To evaluate the expression \(\frac{i}{2-i}\), we will follow these steps:
### Step 1: Multiply by the Conjugate
To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 - i\) is \(2 + i\).
\[
\frac{i}{2 - i} \cdot \frac{2 + i}{2 + i} = \frac{i(2 + i)}{(2 - i)(2 + i)}
\]
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