the points `(alpha,beta) , (gamma , delta) ,(alpha , delta) , (gamma , beta) ` are different real numbers are:

Options

A. Collinear
B. Vertices of a square
C. Vertices of a rhombus
D. Concyclic

Answer

To determine the nature of the points \((\alpha, \beta)\), \((\gamma, \delta)\), \((\alpha, \delta)\), and \((\gamma, \beta)\), we will analyze their properties step by step.

### Step 1: Identify the Points
Let's denote the points as follows:
- Point A: \((\alpha, \beta)\)
- Point B: \((\gamma, \delta)\)
- Point C: \((\alpha, \delta)\)
- Point D: \((\gamma, \beta)\)

### Step 2: Check for Collinearity
To check if the points are collinear, we can use the concept of slopes. If the slopes between any two pairs of points are equal, then the points are collinear.

1. **Slope of AB**:
\[ \text{slope}_{AB} = \frac{\delta - \beta}{\gamma - \alpha} \]

2. **Slope of AC**:
\[ \text{slope}_{AC} = \frac{\delta - \beta}{\alpha - \alpha} = \text{undefined (vertical line)} \]

Since the slope of AC is undefined, it indicates that points A and C are vertically aligned. Therefore, points A, B, C, and D cannot be collinear.