A, B C and D are the points of intersection with the coordinate axes of the lines ax+by=ab and bx+ay=ab, then

Options

A. A, B, C, D are concyclic
B. A, B, C, D form a parallelogram
C. A, B, C, D form a rhombus
D. None of the above

Answer

To solve the problem, we need to find the points of intersection of the lines \( ax + by = ab \) and \( bx + ay = ab \) with the coordinate axes, and then determine the relationship among these points.

### Step-by-Step Solution:

1. **Identify the equations**:
We have two equations:
- Equation 1: \( ax + by = ab \)
- Equation 2: \( bx + ay = ab \)

2. **Find the intersection points with the x-axis**:
To find the intersection points with the x-axis, we set \( y = 0 \).

- For Equation 1:
\[ ax + b(0) = ab \implies ax = ab \implies x = \frac{ab}{a} = b \quad \text{(assuming } a \neq 0\text{)} \]
Thus, the point \( A \) is \( (b, 0) \).

- For Equation 2:
\[ bx + a(0) = ab \implies bx = ab \implies x = \frac{ab}{b} = a \quad \text{(assuming } b \neq 0\text{)} \]
Thus, the point \( C \) is \( (a, 0) \).

3. **Find the intersection points with the y-axis**:
To find the intersection points with the y-axis, we set \( x = 0 \).

- For Equation 1:
\[ a(0) + by = ab \implies by = ab \implies y = \frac{ab}{b} = a \quad \text{(assuming } b \neq 0\text{)} \]
Thus, the point \( B \) is \( (0, a) \).

- For Equation 2:
\[ b(0) + ay = ab \implies ay = ab \implies y = \frac{ab}{a} = b \quad \text{(assuming } a \neq 0\text{)} \]
Thus, the point \( D \) is \( (0, b) \).

4. **List the points**:
We have the points:
- \( A(b, 0) \)
- \( B(0, a) \)
- \( C(a, 0) \)
- \( D(0, b) \)

5. **Check for concyclic points**:
To check if points \( A, B, C, D \) are concyclic, we note that any three points among \( A, B, C, D \) do not lie on a single line. Therefore, these four points can be chosen to form a circle.

### Conclusion:
The points \( A, B, C, D \) are concyclic points.