P, Q, R and S are the points of intersection with the co-ordinate axes of the lines ax+by = ab and bx + ay = ab then

Options

A. P,Q,R,S are concyclic
B. P, Q, R, S form a parallelogram
C. P,Q,R,S form a rhombus
D. none of these

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 check if the points \( P, Q, R, S \) are concyclic.

### Step-by-Step Solution:

1. **Find the intercepts of the line \( ax + by = ab \)**:
- To find the x-intercept (point \( P \)), set \( y = 0 \):
\[ ax + b(0) = ab \implies ax = ab \implies x = b \implies P(b, 0) \]
- To find the y-intercept (point \( Q \)), set \( x = 0 \):
\[ a(0) + by = ab \implies by = ab \implies y = a \implies Q(0, a) \]

2. **Find the intercepts of the line \( bx + ay = ab \)**:
- To find the x-intercept (point \( R \)), set \( y = 0 \):
\[ bx + a(0) = ab \implies bx = ab \implies x = a \implies R(a, 0) \]
- To find the y-intercept (point \( S \)), set \( x = 0 \):
\[ b(0) + ay = ab \implies ay = ab \implies y = b \implies S(0, b) \]

3. **List the points of intersection**:
- The points are:
- \( P(b, 0) \)
- \( Q(0, a) \)
- \( R(a, 0) \)
- \( S(0, b) \)

4. **Check the concyclic condition**:
- For points \( P, Q, R, S \) to be concyclic, the following condition must hold:
\[ OP \cdot OR = OQ \cdot OS \]
- Calculate the distances:
- \( OP = b \)
- \( OQ = a \)
- \( OR = a \)
- \( OS = b \)

5. **Substitute the values into the concyclic condition**:
\[ OP \cdot OR = b \cdot a \]
\[ OQ \cdot OS = a \cdot b \]
- Both sides are equal:
\[ ab = ab \]
- Thus, the condition is satisfied.

### Conclusion:
The points \( P, Q, R, S \) are concyclic.