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

Options

A. `(a^(2)-b^(2))/(ab)`
B. `(b^(2)-a^(2))/(2)`
C. `(b^(2)-a^(2))/(2ab)`
D. none of these

Answer

To find the tangent of 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
The intercept form of a line is given by \( \frac{x}{a} + \frac{y}{b} = 1 \).

For the first line with intercepts \( (a, -b) \):
- The equation of the line is \( \frac{x}{a} + \frac{y}{-b} = 1 \).
- Rearranging gives us \( y = -\frac{b}{a}x + b \).
- The slope \( m_1 \) of the first line is \( -\frac{b}{a} \).

For the second line with intercepts \( (b, -a) \):
- The equation of the line is \( \frac{x}{b} + \frac{y}{-a} = 1 \).
- Rearranging gives us \( y = -\frac{a}{b}x + a \).
- The slope \( m_2 \) of the second line is \( -\frac{a}{b} \).

### Step 2: Use the formula for the tangent of the angle between two lines
The formula for the tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan(\theta) = \frac{m_1 - m_2}{1 + m_1 m_2} \]

### Step 3: Substitute the slopes into the formula
Substituting \( m_1 = -\frac{b}{a} \) and \( m_2 = -\frac{a}{b} \) into the formula:
\[ \tan(\theta) = \frac{-\frac{b}{a} - \left(-\frac{a}{b}\right)}{1 + \left(-\frac{b}{a}\right)\left(-\frac{a}{b}\right)} \]
This simplifies to:
\[ \tan(\theta) = \frac{-\frac{b}{a} + \frac{a}{b}}{1 + \frac{b}{a} \cdot \frac{a}{b}} \]
\[ = \frac{-\frac{b}{a} + \frac{a}{b}}{1 + 1} \]
\[ = \frac{-\frac{b}{a} + \frac{a}{b}}{2} \]

### Step 4: Simplify the expression
Finding a common denominator for the numerator:
\[ = \frac{-\frac{b^2 - a^2}{ab}}{2} \]
\[ = \frac{a^2 - b^2}{2ab} \]

### Final Result
Thus, the tangent of the angle between the lines is:
\[ \tan(\theta) = \frac{b^2 - a^2}{2ab} \]