If z =x +iy lies in the third quadrant then `(barz)/(z)` also lies in third quadrant if (i) `x gt y gt 0` (ii) `x lt y lt 0 ` (iii) `y lt x lt 0` (iv) `y gt x gt 0`

Options

A. `x gt y gt 0`
B. `x lt y lt 0 `
C. `y lt x lt 0`
D. `y gt x gt 0`

Answer

To solve the problem, we need to analyze the conditions under which the complex number \( z = x + iy \) lies in the third quadrant and how the expression \( \frac{\bar{z}}{z} \) behaves.

### Step-by-Step Solution

1. **Identify the Quadrant**:
Since \( z = x + iy \) lies in the third quadrant, we know that both \( x \) and \( y \) are negative. Therefore, we can write:
\[ x < 0 \quad \text{and} \quad y < 0 \]

2. **Calculate \( \bar{z} \)**:
The conjugate of \( z \) is given by:
\[ \bar{z} = x - iy \]

3. **Formulate \( \frac{\bar{z}}{z} \)**:
We can express \( \frac{\bar{z}}{z} \) as follows:
\[ \frac{\bar{z}}{z} = \frac{x - iy}{x + iy} \]

4. **Multiply by the Conjugate**:
To simplify this expression, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{\bar{z}}{z} = \frac{(x - iy)(x - iy)}{(x + iy)(x - iy)} = \frac{x^2 - 2i xy - y^2}{x^2 + y^2} \]

5. **Separate Real and Imaginary Parts**:
This gives us:
\[ \frac{\bar{z}}{z} = \frac{x^2 - y^2}{x^2 + y^2} - i \frac{2xy}{x^2 + y^2} \]
Here, the real part is \( \frac{x^2 - y^2}{x^2 + y^2} \) and the imaginary part is \( -\frac{2xy}{x^2 + y^2} \).

6. **Determine Conditions for Third Quadrant**:
For \( \frac{\bar{z}}{z} \) to lie in the third quadrant, both the real and imaginary parts must be negative:
\[ \frac{x^2 - y^2}{x^2 + y^2} < 0 \quad \text{and} \quad -\frac{2xy}{x^2 + y^2} < 0 \]

7. **Analyze the Real Part**:
The condition \( \frac{x^2 - y^2}{x^2 + y^2} < 0 \) implies:
\[ x^2 < y^2 \quad \Rightarrow \quad |x| < |y| \]
Since both \( x \) and \( y \) are negative, this translates to:
\[ x < y < 0 \]

8. **Analyze the Imaginary Part**:
The condition \( -\frac{2xy}{x^2 + y^2} < 0 \) implies:
\[ xy > 0 \]
Since both \( x \) and \( y \) are negative, this condition is satisfied.

9. **Conclusion**:
The conditions derived show that for \( \frac{\bar{z}}{z} \) to lie in the third quadrant, we must have:
\[ x < y < 0 \]
Thus, the correct option is:
\[ \text{(ii) } x < y < 0 \]