If the lines `y = 3x + 1` and `2y = x + 3` are equally inclined to the line `y=mx +4 , (1/2 < m < 3)` then find the values `m`

Options

A. `(1+3sqrt(2))/(7)`
B. `(1-3sqrt(2))/(7)`
C. `(1+-3sqrt(2))/(7)`
D. `(1+-5sqrt(2))/(7)`

Explanation

To solve the problem, we need to find the values of \( m \) such that the lines \( y = 3x + 1 \) and \( 2y = x + 3 \) are equally inclined to the line \( y = mx + 4 \).

### Step-by-Step Solution:

1. **Identify the slopes of the lines:**
- The slope of the first line \( y = 3x + 1 \) is \( m_1 = 3 \).
- The slope of the second line \( 2y = x + 3 \) can be rewritten as \( y = \frac{1}{2}x + \frac{3}{2} \), so its slope is \( m_2 = \frac{1}{2} \).
- The slope of the third line \( y = mx + 4 \) is \( m \).

2. **Use the formula for the tangent of the angle between two lines:**
- The tangent of the angle \( \theta_1 \) between the first line and the third line is given by:
\[ \tan \theta_1 = \frac{m_1 - m}{1 + m_1 m} = \frac{3 - m}{1 + 3m} \]
- The tangent of the angle \( \theta_2 \) between the second line and the third line is given by:
\[ \tan \theta_2 = \frac{m - m_2}{1 + m m_2} = \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} = \frac{2m - 1}{2 + m} \]

3. **Set the angles equal since they are equally inclined:**
- Since \( \theta_1 = \theta_2 \), we can set the two expressions for tangent equal to each other:
\[ \frac{3 - m}{1 + 3m} = \frac{2m - 1}{2 + m} \]

4. **Cross-multiply to eliminate the fractions:**
- Cross-multiplying gives us:
\[ (3 - m)(2 + m) = (2m - 1)(1 + 3m) \]

5. **Expand both sides:**
- Expanding the left side:
\[ 6 + 3m - 2m - m^2 = 6 + m - m^2 \]
- Expanding the right side:
\[ 2m + 6m^2 - 1 - 3m = 6m^2 - m - 1 \]

6. **Combine like terms:**
- Setting both sides equal:
\[ 6 + m - m^2 = 6m^2 - m - 1 \]
- Rearranging gives:
\[ 7m^2 - 2m - 7 = 0 \]

7. **Solve the quadratic equation:**
- Using the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 7, b = -2, c = -7 \).
\[ m = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 7 \cdot (-7)}}{2 \cdot 7} \]
\[ m = \frac{2 \pm \sqrt{4 + 196}}{14} \]
\[ m = \frac{2 \pm \sqrt{200}}{14} \]
\[ m = \frac{2 \pm 10\sqrt{2}}{14} \]
\[ m = \frac{1 \pm 5\sqrt{2}}{7} \]

8. **Determine valid values of \( m \):**
- We have two potential values:
\[ m_1 = \frac{1 + 5\sqrt{2}}{7}, \quad m_2 = \frac{1 - 5\sqrt{2}}{7} \]
- Since \( \frac{1}{2} < m < 3 \), we need to evaluate both values to see which one(s) fit this range.