Area of the triangle with vertces (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. `ab(r-1)(s-1)(s-r)`
B. `1/2ab(r+1)(s+1)(s-r)`
C. `1/2ab(r-1)(s-1)(s-r)`
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 Geometric Progression (G.P.) with common ratio \( r \) and \( b, y_1, y_2 \) are in G.P. with common ratio \( s \), we can follow these steps:

### Step 1: Express the coordinates in terms of \( a \) and \( b \)

Since \( a, x_1, x_2 \) are in G.P. with common ratio \( r \), we can express \( x_1 \) and \( x_2 \) as:
- \( x_1 = a \cdot r \)
- \( x_2 = a \cdot r^2 \)

Similarly, since \( b, y_1, y_2 \) are in G.P. with common ratio \( s \), we can express \( y_1 \) and \( y_2 \) as:
- \( y_1 = b \cdot s \)
- \( y_2 = b \cdot s^2 \)

### Step 2: Use the formula for the area of a triangle

The area \( A \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our vertices \( (a, b) \), \( (x_1, y_1) \), and \( (x_2, y_2) \), we substitute:
- \( (x_1, y_1) = (ar, bs) \)
- \( (x_2, y_2) = (ar^2, bs^2) \)
- \( (x_3, y_3) = (a, b) \)

### Step 3: Substitute the coordinates into the area formula

Substituting the coordinates into the area formula:
\[ A = \frac{1}{2} \left| a(bs^2 - b) + ar(bs - b) + a(ar^2 - ar) \right| \]

### Step 4: Simplify the expression

Now, simplifying the expression:
\[ A = \frac{1}{2} \left| ab(s^2 - 1) + ar(b(s - 1)) + a^2r^2(r - 1) \right| \]
\[ = \frac{1}{2} \left| ab(s^2 - 1) + arb(s - 1) + a^2r^2(r - 1) \right| \]

### Step 5: Final expression for the area

Thus, the area of the triangle is:
\[ A = \frac{1}{2} \left| a \cdot b \cdot (s^2 - 1) + a \cdot r \cdot b \cdot (s - 1) + a^2 \cdot r^2 \cdot (r - 1) \right| \]