Key Concepts and Formulas

• Geometric Progression (G.P.): A sequence of numbers where each term is multiplied by a constant value (common ratio) to get the next term. A G.P. can be represented as $a, ar, ar^2, ar^3, ...$, where $a$ is the first term and $r$ is the common ratio.
• Collinearity of Three Points: Three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are collinear if and only if the slope between any two pairs of points is the same. Mathematically, this means $\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2}$.
• Area of a Triangle: The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$ The points are collinear if and only if the area of the triangle formed by them is zero.

Step-by-Step Solution

Step 1: Define the points using the properties of G.P. Since $x_1, x_2, x_3$ are in G.P. with common ratio $r$, we can write: $x_2 = x_1r$ and $x_3 = x_1r^2$.

Similarly, since $y_1, y_2, y_3$ are in G.P. with the same common ratio $r$, we can write: $y_2 = y_1r$ and $y_3 = y_1r^2$.

Therefore, the points are $(x_1, y_1)$, $(x_1r, y_1r)$, and $(x_1r^2, y_1r^2)$.

Step 2: Check for collinearity using the slope condition. We will calculate the slope between the first two points and the slope between the second and third points. If the slopes are equal, the points are collinear.

Slope between $(x_1, y_1)$ and $(x_1r, y_1r)$: $m_1 = \frac{y_1r - y_1}{x_1r - x_1} = \frac{y_1(r - 1)}{x_1(r - 1)}$ If $r \neq 1$ and $x_1 \neq 0$, then $m_1 = \frac{y_1}{x_1}$.

Slope between $(x_1r, y_1r)$ and $(x_1r^2, y_1r^2)$: $m_2 = \frac{y_1r^2 - y_1r}{x_1r^2 - x_1r} = \frac{y_1r(r - 1)}{x_1r(r - 1)}$ If $r \neq 1$ and $x_1 \neq 0$ and $r \neq 0$, then $m_2 = \frac{y_1}{x_1}$.

Since $m_1 = m_2 = \frac{y_1}{x_1}$, the points are collinear (lie on a straight line), provided $x_1 \neq 0$ and $r \neq 0$ and $r \neq 1$.

Step 3: Consider the case when $r=1$. If $r = 1$, then the points are $(x_1, y_1), (x_1, y_1), (x_1, y_1)$, which are trivially collinear.

Step 4: Consider the case when $x_1=0$. If $x_1 = 0$, then $x_2 = x_3 = 0$, and the points are $(0, y_1), (0, y_1r), (0, y_1r^2)$, which lie on the y-axis (x=0) and are collinear.

Step 5: Consider the case when $r=0$. If $r = 0$, then $x_2 = x_3 = 0$ and $y_2 = y_3 = 0$. The points are $(x_1, y_1), (0, 0), (0, 0)$, which are collinear.

Common Mistakes & Tips

• Always remember to consider the special cases when the common ratio $r=0$ or $r=1$, or when the first term is zero.
• Using the determinant form of the area of a triangle is another way to check for collinearity. If the determinant is zero, the points are collinear.
• Understanding the definition of a geometric progression is crucial for solving this problem.

Summary

The problem states that $x_1, x_2, x_3$ and $y_1, y_2, y_3$ are in G.P. with the same common ratio. By expressing the points in terms of the first terms $x_1, y_1$ and the common ratio $r$, and then calculating the slopes between pairs of points, we found that the slopes are equal, implying that the points are collinear. The analysis holds true even for special cases like $r=0, r=1$ and $x_1=0$.

The final answer is \boxed{B}, which corresponds to option (B).