Straight lines `(x)/(a)+(y)/(b)=1, (x)/(b)+(y)/(a)=1,(x)/(a)+(y)/(b)=2 and (x)/(b)+(y)/(a)=2` form a rhombus of area ( in square units)

Options

A. `(ab)/(|a^(2)-b^(2)|)`
B. `(ab)/(a^(2)+b^(2))`
C. `(a^(2)b^(2))/(a^(2)+b^(2))`
D. `(a^(2)b^(2))/(|a^(2)-b^(2)|)`

Explanation

To find the area of the rhombus formed by the given lines, we will follow these steps:

### Step 1: Identify the equations of the lines
The given lines are:
1. \( \frac{x}{a} + \frac{y}{b} = 1 \)
2. \( \frac{x}{b} + \frac{y}{a} = 1 \)
3. \( \frac{x}{a} + \frac{y}{b} = 2 \)
4. \( \frac{x}{b} + \frac{y}{a} = 2 \)

### Step 2: Convert the equations to standard form
We can rewrite the equations in standard form \( Ax + By = C \):
1. \( x + \frac{b}{a}y = b \) (from \( \frac{x}{a} + \frac{y}{b} = 1 \))
2. \( \frac{1}{b}x + \frac{1}{a}y = 1 \) (from \( \frac{x}{b} + \frac{y}{a} = 1 \))
3. \( x + \frac{b}{a}y = 2b \) (from \( \frac{x}{a} + \frac{y}{b} = 2 \))
4. \( \frac{1}{b}x + \frac{1}{a}y = 2 \) (from \( \frac{x}{b} + \frac{y}{a} = 2 \))

### Step 3: Identify the slopes of the lines
From the equations, we can find the slopes:
- For lines 1 and 3: The slope \( m_1 = -\frac{b}{a} \)
- For lines 2 and 4: The slope \( m_2 = -\frac{a}{b} \)

### Step 4: Find the distance between the parallel lines
The distance \( d \) between two parallel lines of the form \( Ax + By = C_1 \) and \( Ax + By = C_2 \) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

**For the first set of lines:**
- \( C_1 = b \) and \( C_2 = 2b \)
- \( A = 1, B = \frac{b}{a} \)

Calculating the distance:
\[ d_1 = \frac{|2b - b|}{\sqrt{1^2 + \left(\frac{b}{a}\right)^2}} = \frac{b}{\sqrt{1 + \frac{b^2}{a^2}}} = \frac{b}{\sqrt{\frac{a^2 + b^2}{a^2}}} = \frac{ab}{\sqrt{a^2 + b^2}} \]

**For the second set of lines:**
- \( C_1 = 1 \) and \( C_2 = 2 \)
- \( A = \frac{1}{b}, B = \frac{1}{a} \)

Calculating the distance:
\[ d_2 = \frac{|2 - 1|}{\sqrt{\left(\frac{1}{b}\right)^2 + \left(\frac{1}{a}\right)^2}} = \frac{1}{\sqrt{\frac{1}{b^2} + \frac{1}{a^2}}} = \frac{1}{\frac{\sqrt{a^2 + b^2}}{ab}} = \frac{ab}{\sqrt{a^2 + b^2}} \]

### Step 5: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Substituting the distances:
\[ A = \frac{1}{2} \times \frac{ab}{\sqrt{a^2 + b^2}} \times \frac{ab}{\sqrt{a^2 + b^2}} = \frac{1}{2} \times \frac{a^2b^2}{a^2 + b^2} \]

### Final Answer
The area of the rhombus is:
\[ \frac{a^2b^2}{2(a^2 + b^2)} \text{ square units} \]