The centre of a square is at the origin and 1 + i is one of its vertices. The extremities of its diagonal which does not pass 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 find the extremities of the diagonal of a square that does not pass through the vertex \(1 + i\), we can follow these steps:

### Step 1: Identify the center and vertex
The center of the square is given as the origin, which is \(0 + 0i\). One of the vertices is \(1 + i\).

### Step 2: Calculate the distance from the center to the vertex
Since the center is at the origin, the distance from the center to the vertex \(1 + i\) can be calculated using the modulus:
\[ d = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 3: Determine the other vertices of the square
The vertices of the square can be determined by rotating the vertex \(1 + i\) around the center (the origin) by \(90^\circ\) and \(270^\circ\).

1. **Rotation by \(90^\circ\)**:
- The vertex \(1 + i\) can be rotated to:
\[ (1 + i) \cdot e^{i\frac{\pi}{2}} = (1 + i)(0 + i) = -1 + i \]

2. **Rotation by \(270^\circ\)**:
- The vertex \(1 + i\) can be rotated to:
\[ (1 + i) \cdot e^{-i\frac{\pi}{2}} = (1 + i)(0 - i) = 1 - i \]

### Step 4: Identify the extremities of the diagonal that does not pass through \(1 + i\)
The vertices of the square are:
- \(1 + i\) (given)
- \(-1 - i\) (opposite vertex)
- \(-1 + i\) (90-degree rotation)
- \(1 - i\) (270-degree rotation)

The diagonal that does not pass through \(1 + i\) connects the points:
- \(-1 - i\) and \(1 - i\)

### Conclusion
The extremities of the diagonal that does not pass through the vertex \(1 + i\) are:
\[ \text{Extremities: } -1 - i \text{ and } 1 - i \]