If `x_1, x_2, x_3` as well as `y_1, y_2, y_3` are in GP with same common ratio, then the points `P(x_1, y_1), Q(x_2, y_2)` and `R(x_3, y_3)`

Options

A. lie on a straight line
B. lie on an ellipse
C. lie on a circle
D. are vertices of a triangle

Answer

To solve the problem, we need to determine the relationship between the points \( P(x_1, y_1) \), \( Q(x_2, y_2) \), and \( R(x_3, y_3) \) given that \( x_1, x_2, x_3 \) and \( y_1, y_2, y_3 \) are in geometric progression (GP) with the same common ratio.

### Step-by-Step Solution:

1. **Understanding the Geometric Progression**:
Since \( x_1, x_2, x_3 \) are in GP, we can express them in terms of \( x_1 \) and a common ratio \( r \):
\[ x_2 = x_1 \cdot r \quad \text{and} \quad x_3 = x_1 \cdot r^2 \]
Similarly, for \( y_1, y_2, y_3 \):
\[ y_2 = y_1 \cdot r \quad \text{and} \quad y_3 = y_1 \cdot r^2 \]

2. **Setting Up the Determinant**:
To check if the points \( P, Q, R \) are collinear, we can use the area of the triangle formed by these points. The area can be calculated using the determinant:
\[ \text{Area} = \frac{1}{2} \begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \end{vmatrix} \]

3. **Substituting the Values**:
Substitute \( x_2 \) and \( x_3 \) as well as \( y_2 \) and \( y_3 \) into the determinant:
\[ \begin{vmatrix} x_1 & x_1 r & x_1 r^2 \\ y_1 & y_1 r & y_1 r^2 \\ 1 & 1 & 1 \end{vmatrix} \]

4. **Factoring Out Common Terms**:
Factor out \( x_1 \) from the first column and \( y_1 \) from the second column:
\[ = x_1 y_1 \begin{vmatrix} 1 & r & r^2 \\ 1 & r & r^2 \\ 1 & 1 & 1 \end{vmatrix} \]

5. **Evaluating the Determinant**:
Notice that the first two rows of the determinant are identical:
\[ \begin{vmatrix} 1 & r & r^2 \\ 1 & r & r^2 \\ 1 & 1 & 1 \end{vmatrix} = 0 \]
Since two rows are the same, the determinant is zero.

6. **Conclusion**:
Since the area is zero, the points \( P, Q, R \) are collinear. Therefore, they lie on a straight line.

### Final Answer:
The points \( P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3) \) are collinear and lie on a straight line.

---