If the straight line `a x+c y=2b ,` where `a , b , c >0,` makes a triangle of area 2 sq. units with the coordinate axes, then (a) `a , b , c` are in GP (b) a, -b, c are in GP (c) `a ,2b ,c` are in GP (d) `a ,-2b ,c` are in GP

Options

A. a,b,c are in GP
B. a,-b, c are in GP
C. a,2b,c are in GP
D. a,-2b, c are in GP

Explanation

To solve the problem step by step, we need to find the relationship between the constants \(a\), \(b\), and \(c\) given that the line \(ax + cy = 2b\) forms a triangle with the coordinate axes that has an area of 2 square units.

### Step 1: Find the intercepts of the line with the axes
The line \(ax + cy = 2b\) can be rewritten to find the x-intercept and y-intercept.

- **X-intercept**: Set \(y = 0\):
\[ ax + c(0) = 2b \implies ax = 2b \implies x = \frac{2b}{a} \]
So, the x-intercept is \(\left(\frac{2b}{a}, 0\right)\).

- **Y-intercept**: Set \(x = 0\):
\[ a(0) + cy = 2b \implies cy = 2b \implies y = \frac{2b}{c} \]
So, the y-intercept is \(\left(0, \frac{2b}{c}\right)\).

### Step 2: Calculate the area of the triangle formed by the intercepts
The area \(A\) of a triangle formed by the x-intercept and y-intercept can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In our case, the base is \(\frac{2b}{a}\) and the height is \(\frac{2b}{c}\):
\[ A = \frac{1}{2} \times \frac{2b}{a} \times \frac{2b}{c} = \frac{2b^2}{ac} \]

### Step 3: Set the area equal to 2 square units
According to the problem, the area of the triangle is given as 2 square units:
\[ \frac{2b^2}{ac} = 2 \]

### Step 4: Simplify the equation
To simplify, we can multiply both sides by \(ac\):
\[ 2b^2 = 2ac \]
Dividing both sides by 2 gives:
\[ b^2 = ac \]

### Step 5: Determine the relationship between \(a\), \(b\), and \(c\)
The equation \(b^2 = ac\) indicates that \(a\), \(b\), and \(c\) are in a geometric progression (GP). This is because in a GP, the square of the middle term is equal to the product of the other two terms.

### Conclusion
Thus, we can conclude that:
- \(a\), \(b\), and \(c\) are in GP.