The angle between the lines whose intercepts on the axes are a, –b and b, –a respectively , is

Options

A. `tan ^(-1) "" (a ^(2) -b ^(2))/( ab )`
B. `tan ^(-1) "" ( b ^(2) -a ^(2))/( 2)`
C. `tan ^(-1) "" ( b ^(2) -a ^(2))/( 2 ab) `
D. None of these

Answer

To find the angle between the lines whose intercepts on the axes are \( (a, -b) \) and \( (b, -a) \), we can follow these steps:

### Step 1: Determine the slopes of the lines

1. **Line 1** has intercepts at \( (a, 0) \) and \( (0, -b) \).
- The slope \( m_1 \) of the line can be calculated using the formula:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-b - 0}{0 - a} = \frac{-b}{-a} = \frac{b}{a} \]

2. **Line 2** has intercepts at \( (b, 0) \) and \( (0, -a) \).
- The slope \( m_2 \) of the line can be calculated similarly:
\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-a - 0}{0 - b} = \frac{-a}{-b} = \frac{a}{b} \]

### Step 2: Use the formula for the angle between two lines

The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the slopes into the formula

Substituting \( m_1 = \frac{b}{a} \) and \( m_2 = \frac{a}{b} \) into the formula:
\[ \tan \theta = \left| \frac{\frac{b}{a} - \frac{a}{b}}{1 + \frac{b}{a} \cdot \frac{a}{b}} \right| \]

### Step 4: Simplify the expression

1. The numerator becomes:
\[ \frac{b}{a} - \frac{a}{b} = \frac{b^2 - a^2}{ab} \]

2. The denominator simplifies to:
\[ 1 + \frac{b}{a} \cdot \frac{a}{b} = 1 + 1 = 2 \]

Thus, we have:
\[ \tan \theta = \left| \frac{\frac{b^2 - a^2}{ab}}{2} \right| = \frac{b^2 - a^2}{2ab} \]

### Step 5: Find the angle \( \theta \)

Finally, we can express the angle \( \theta \) as:
\[ \theta = \tan^{-1} \left( \frac{b^2 - a^2}{2ab} \right) \]

### Final Result

The angle between the lines whose intercepts on the axes are \( (a, -b) \) and \( (b, -a) \) is:
\[ \theta = \tan^{-1} \left( \frac{b^2 - a^2}{2ab} \right) \]