Find the equations of the lines which pass through the origin and are inclined at an angle `tan^(-1)m` to the line `y=m x+c dot`

Explanation

To find the equations of the lines that pass through the origin and are inclined at an angle \( \tan^{-1} m \) to the line \( y = mx + c \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the slope of the given line**:
The line given is \( y = mx + c \). The slope of this line is \( m \).

2. **Determine the angle of inclination**:
The angle \( \theta \) that this line makes with the x-axis can be expressed as:
\[ \theta = \tan^{-1}(m) \]

3. **Calculate the angle for the new lines**:
We need to find lines that make an angle of \( \tan^{-1}(m) \) with the line \( y = mx + c \). This means the new lines will make angles of \( \theta + \tan^{-1}(m) \) and \( \theta - \tan^{-1}(m) \) with the x-axis.

4. **Use the tangent addition formula**:
For the angle \( \theta + \tan^{-1}(m) \), we can use the tangent addition formula:
\[ \tan(\theta + \tan^{-1}(m)) = \frac{\tan(\theta) + \tan(\tan^{-1}(m))}{1 - \tan(\theta) \tan(\tan^{-1}(m))} \]
Substituting \( \tan(\theta) = m \) and \( \tan(\tan^{-1}(m)) = m \):
\[ \tan(\theta + \tan^{-1}(m)) = \frac{m + m}{1 - m^2} = \frac{2m}{1 - m^2} \]

5. **Equation of the first line**:
The slope of the first line (let's call it \( L_1 \)) is \( \frac{2m}{1 - m^2} \). Since it passes through the origin, its equation is:
\[ y = \frac{2m}{1 - m^2} x \]

6. **Calculate the angle for the second line**:
For the angle \( \theta - \tan^{-1}(m) \), we again use the tangent subtraction formula:
\[ \tan(\theta - \tan^{-1}(m)) = \frac{\tan(\theta) - \tan(\tan^{-1}(m))}{1 + \tan(\theta) \tan(\tan^{-1}(m))} \]
Substituting \( \tan(\theta) = m \):
\[ \tan(\theta - \tan^{-1}(m)) = \frac{m - m}{1 + m^2} = 0 \]

7. **Equation of the second line**:
The slope of the second line (let's call it \( L_2 \)) is \( 0 \). Thus, the equation of this line is:
\[ y = 0 \]

### Final Answer:
The equations of the lines that pass through the origin and are inclined at an angle \( \tan^{-1}(m) \) to the line \( y = mx + c \) are:
1. \( y = \frac{2m}{1 - m^2} x \)
2. \( y = 0 \)