The centre of a square is at the origin and one of the vertex is `1-i` extremities of diagonal not passing through this vertex are

Options

A. `1-I, -1+i`
B. `1-I,-1-i`
C. `-1+I, -1-i`
D. none of these

Answer

To solve the problem, we need to find the extremities of the diagonal of a square that does not pass through the vertex given as \(1 - i\), with the center of the square at the origin.

### Step-by-Step Solution:

1. **Identify the Given Vertex**:
The vertex of the square is given as \( z_1 = 1 - i \).

2. **Determine the Center**:
The center of the square is at the origin, which can be represented as \( z_c = 0 + 0i \).

3. **Find the Length of the Diagonal**:
The length of the diagonal of the square can be calculated using the distance from the center to the vertex. The distance \( d \) from the center to the vertex \( z_1 \) is:
\[ d = |z_1 - z_c| = |(1 - i) - 0| = |1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{2} \]
Since the diagonal of a square is \( d \), the vertices of the square will be at a distance of \( \frac{d}{\sqrt{2}} \) from the center.

4. **Calculate the Other Vertices**:
The other vertices can be found by rotating the vertex \( z_1 \) by \( 90^\circ \) (or \( \frac{\pi}{2} \) radians) in both clockwise and counterclockwise directions.

5. **Rotation by \( 90^\circ \) Counterclockwise**:
The formula for rotation in the complex plane is:
\[ z_2 = z_1 \cdot e^{i\frac{\pi}{2}} = (1 - i) \cdot i \]
Calculating this:
\[ z_2 = (1 - i) \cdot i = i - i^2 = i + 1 = 1 + i \]

6. **Rotation by \( 90^\circ \) Clockwise**:
For clockwise rotation, we use:
\[ z_3 = z_1 \cdot e^{-i\frac{\pi}{2}} = (1 - i) \cdot (-i) \]
Calculating this:
\[ z_3 = (1 - i)(-i) = -i + i^2 = -i - 1 = -1 - i \]

7. **Final Result**:
The extremities of the diagonal not passing through the vertex \( 1 - i \) are:
\[ z_2 = 1 + i \quad \text{and} \quad z_3 = -1 - i \]

### Summary:
The extremities of the diagonal not passing through the vertex \( 1 - i \) are \( 1 + i \) and \( -1 - i \).