If P, Q, R, S are four coplanar points on the sides AB, BC, CD, DA of a skew quadrilateral, then `(AB)/(PB)cdot(BQ)/(QC)cdot(CR)/(RD)cdot(DS)/(SA)` equals

Options

A. `1`
B. `-1`
C. `3`
D. `-3`

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{AB}{PB} \cdot \frac{BQ}{QC} \cdot \frac{CR}{RD} \cdot \frac{DS}{SA} \]

where \( P, Q, R, S \) are coplanar points on the sides \( AB, BC, CD, DA \) of a skew quadrilateral \( ABCD \).

### Step-by-Step Solution:

1. **Understanding the Ratios**:
- Let \( P \) divide \( AB \) in the ratio \( \alpha : 1 \).
- Let \( Q \) divide \( BC \) in the ratio \( \beta : 1 \).
- Let \( R \) divide \( CD \) in the ratio \( \gamma : 1 \).
- Let \( S \) divide \( DA \) in the ratio \( \delta : 1 \).

2. **Expressing the Coordinates**:
- The coordinates of point \( P \) can be expressed as:
\[ P = \left( \frac{\alpha x_2 + x_1}{\alpha + 1}, \frac{\alpha y_2 + y_1}{\alpha + 1}, \frac{\alpha z_2 + z_1}{\alpha + 1} \right) \]
- Similarly, we can express the coordinates for points \( Q, R, S \) using their respective ratios.

3. **Setting up the Plane Equation**:
- Since \( P, Q, R, S \) are coplanar, they must satisfy the plane equation. The general form of the plane equation is:
\[ ax + by + cz + d = 0 \]
- Substitute the coordinates of points \( P, Q, R, S \) into this equation to derive relationships between \( \alpha, \beta, \gamma, \delta \).

4. **Finding the Values of Ratios**:
- After substituting, we will find:
\[ \alpha = -\frac{p_1}{p_2}, \quad \beta = -\frac{p_2}{p_3}, \quad \gamma = -\frac{p_3}{p_4}, \quad \delta = -\frac{p_4}{p_1} \]
- Here, \( p_1, p_2, p_3, p_4 \) are the determinants formed from the coordinates of points \( A, B, C, D \).

5. **Calculating the Product**:
- Now, we multiply the ratios:
\[ \alpha \cdot \beta \cdot \gamma \cdot \delta = \left(-\frac{p_1}{p_2}\right) \cdot \left(-\frac{p_2}{p_3}\right) \cdot \left(-\frac{p_3}{p_4}\right) \cdot \left(-\frac{p_4}{p_1}\right) \]
- This simplifies to:
\[ \frac{p_1 p_2 p_3 p_4}{p_2 p_3 p_4 p_1} = 1 \]

6. **Conclusion**:
- Therefore, the value of the expression is:
\[ \frac{AB}{PB} \cdot \frac{BQ}{QC} \cdot \frac{CR}{RD} \cdot \frac{DS}{SA} = 1 \]