If `x_(1), x_(2), x_(3)` as well as `y_(1), y_(2), y_(3)` are in GP, with the same common ratio, then the points `(x_(1),y_(1)), (x_(2),y_(2))` and `(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 determine whether the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) lie on a straight line, a circle, an ellipse, or form the vertices of a triangle, we start by analyzing the given information.

### Step 1: Define the terms
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 \(r\), we can express them as follows:
- Let \(x_1 = a\)
- Then \(x_2 = ar\)
- And \(x_3 = ar^2\)

Similarly, for \(y\):
- Let \(y_1 = b\)
- Then \(y_2 = br\)
- And \(y_3 = br^2\)

### Step 2: Write the points
The points can now be represented as:
- \(P_1 = (x_1, y_1) = (a, b)\)
- \(P_2 = (x_2, y_2) = (ar, br)\)
- \(P_3 = (x_3, y_3) = (ar^2, br^2)\)

### Step 3: Calculate the slopes
To check if the points are collinear, we can calculate the slopes between the points.

#### Slope between \(P_1\) and \(P_2\):
\[ \text{slope}_{P_1P_2} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{br - b}{ar - a} = \frac{b(r - 1)}{a(r - 1)} = \frac{b}{a} \]

#### Slope between \(P_2\) and \(P_3\):
\[ \text{slope}_{P_2P_3} = \frac{y_3 - y_2}{x_3 - x_2} = \frac{br^2 - br}{ar^2 - ar} = \frac{br(r - 1)}{ar(r - 1)} = \frac{b}{a} \]

### Step 4: Compare the slopes
Since both slopes are equal:
\[ \text{slope}_{P_1P_2} = \text{slope}_{P_2P_3} = \frac{b}{a} \]
This indicates that the points \(P_1\), \(P_2\), and \(P_3\) are collinear.

### Conclusion
Since the points are collinear, they lie on a straight line. Therefore, the correct option is:
- **Option 1: Lies on a straight line.**