If ` z_(1)=2-i, z_(2)=1+ 2i,` then find the value of the following :
(i) `Re((z_(1)*z_(2))/(bar(z)_(2)))`
(ii) ` Im (z_(1)*bar(z)_(2))`

Explanation

To solve the given problem step by step, we will find the required values for both parts (i) and (ii).

### Given:
- \( z_1 = 2 - i \)
- \( z_2 = 1 + 2i \)

### Part (i): Find \( \text{Re}\left(\frac{z_1 z_2}{\overline{z_2}}\right) \)

**Step 1:** Find the conjugate of \( z_2 \).
\[ \overline{z_2} = 1 - 2i \]

**Step 2:** Compute the product \( z_1 z_2 \).
\[ z_1 z_2 = (2 - i)(1 + 2i) \]
Using the distributive property:
\[ = 2 \cdot 1 + 2 \cdot 2i - i \cdot 1 - i \cdot 2i \]
\[ = 2 + 4i - i + 2 \]
\[ = 4 + 3i \]

**Step 3:** Now compute \( \frac{z_1 z_2}{\overline{z_2}} \).
\[ \frac{z_1 z_2}{\overline{z_2}} = \frac{4 + 3i}{1 - 2i} \]

**Step 4:** Multiply the numerator and denominator by the conjugate of the denominator.
\[ = \frac{(4 + 3i)(1 + 2i)}{(1 - 2i)(1 + 2i)} \]
Calculating the denominator:
\[ (1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5 \]

Calculating the numerator:
\[ (4 + 3i)(1 + 2i) = 4 \cdot 1 + 4 \cdot 2i + 3i \cdot 1 + 3i \cdot 2i \]
\[ = 4 + 8i + 3i - 6 = -2 + 11i \]

**Step 5:** Now we have:
\[ \frac{z_1 z_2}{\overline{z_2}} = \frac{-2 + 11i}{5} = -\frac{2}{5} + \frac{11}{5}i \]

**Step 6:** Find the real part.
\[ \text{Re}\left(\frac{z_1 z_2}{\overline{z_2}}\right) = -\frac{2}{5} \]

### Part (ii): Find \( \text{Im}(z_1 \overline{z_2}) \)

**Step 1:** Compute \( z_1 \overline{z_2} \).
\[ z_1 \overline{z_2} = (2 - i)(1 - 2i) \]
Using the distributive property:
\[ = 2 \cdot 1 - 2 \cdot 2i - i \cdot 1 + i \cdot 2i \]
\[ = 2 - 4i - i - 2 \]
\[ = 0 - 5i \]

**Step 2:** Find the imaginary part.
\[ \text{Im}(z_1 \overline{z_2}) = -5 \]

### Final Answers:
(i) \( \text{Re}\left(\frac{z_1 z_2}{\overline{z_2}}\right) = -\frac{2}{5} \)

(ii) \( \text{Im}(z_1 \overline{z_2}) = -5 \)

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