Area of the triangle with vertices `(a, b), (x_1,y_1) and (x_2, y_2)` where `a, x_1,x_2` are in G.P. with common ratio `r` and `b, y_1, y_2`, are in G.P with common ratio s, is

Options

A. (a) `ab (r - 1)(s-1) (s-r)`
B. (b) `(1)/(2)ab (r+1) (s+1) (s-r)`
C. (c) `(1)/(2)ab (r-1) (s-1) (s-r)`
D. (d) `ab (r+1) (s+1) (r-s)`

Answer

To find the area of the triangle with vertices \((a, b)\), \((x_1, y_1)\), and \((x_2, y_2)\) where \(a, x_1, x_2\) are in a geometric progression (G.P.) with common ratio \(r\) and \(b, y_1, y_2\) are in a G.P. with common ratio \(s\), we can follow these steps:

### Step 1: Express \(x_1\) and \(x_2\) in terms of \(a\)
Since \(a, x_1, x_2\) are in G.P. with common ratio \(r\):
- \(x_1 = ar\)
- \(x_2 = ar^2\)

### Step 2: Express \(y_1\) and \(y_2\) in terms of \(b\)
Since \(b, y_1, y_2\) are in G.P. with common ratio \(s\):
- \(y_1 = sb\)
- \(y_2 = s^2b\)

### Step 3: Set up the formula for the area of the triangle
The area \(A\) of a triangle formed by the points \((x_1, y_1)\), \((x_2, y_2)\), and \((a, b)\) can be calculated using the determinant formula:
\[ A = \frac{1}{2} \left| \begin{vmatrix} a & b & 1 \\ ar & sb & 1 \\ ar^2 & s^2b & 1 \end{vmatrix} \right| \]

### Step 4: Calculate the determinant
Calculating the determinant:
\[ \begin{vmatrix} a & b & 1 \\ ar & sb & 1 \\ ar^2 & s^2b & 1 \end{vmatrix} \]

Using properties of determinants, we can simplify this by factoring out common terms:
1. Factor \(a\) from the first column and \(b\) from the second column:
\[ = ab \begin{vmatrix} 1 & 1 & 1 \\ r & s & 1 \\ r^2 & s^2 & 1 \end{vmatrix} \]

### Step 5: Perform row operations to simplify the determinant
Now, perform column operations:
- \(C_1 \rightarrow C_1 - C_2\)
- \(C_3 \rightarrow C_3 - C_2\)

This results in:
\[ = ab \begin{vmatrix} 0 & 1 & 0 \\ r - s & 1 & 0 \\ r^2 - s^2 & 1 & 0 \end{vmatrix} \]

### Step 6: Expand the determinant
Now, expand the determinant:
\[ = ab \cdot 0 - (r - s) \cdot 0 + (r^2 - s^2)(1) \]
This simplifies to:
\[ = ab (r - s)(1) \]

### Step 7: Final area calculation
Thus, the area of the triangle is:
\[ A = \frac{1}{2} ab (r - s) \]

### Final Result
The area of the triangle is:
\[ \frac{1}{2} ab (r - s) \]