The angle between `y=x+6` and `y=sqrt(3)x+y` is

Options

A. `15^(@)`
B. `12^(@)`
C. `13^(@)`
D. `14^(@)`

Answer

To find the angle between the two lines given by the equations \(y = x + 6\) and \(y = \sqrt{3}x + y\), we can follow these steps:

### Step 1: Identify the slopes of the lines
The first line is given by:
\[ y = x + 6 \]
This can be rewritten in slope-intercept form \(y = mx + c\), where \(m\) is the slope. Here, the slope \(m_1\) is:
\[ m_1 = 1 \]

The second line is given by:
\[ y = \sqrt{3}x + y \]
This seems to have a typo. Assuming it should be \(y = \sqrt{3}x + c\), we can take the slope \(m_2\) as:
\[ m_2 = \sqrt{3} \]

### 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 the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the values of the slopes
Substituting \(m_1 = 1\) and \(m_2 = \sqrt{3}\) into the formula:
\[ \tan \theta = \left| \frac{1 - \sqrt{3}}{1 + 1 \cdot \sqrt{3}} \right| \]

### Step 4: Simplify the expression
Calculating the numerator and denominator:
\[ \tan \theta = \left| \frac{1 - \sqrt{3}}{1 + \sqrt{3}} \right| \]

### Step 5: Calculate the angle
To find \(\theta\), we can calculate the arctangent:
1. First, calculate the value of \(\tan \theta\).
2. Then, use a calculator or trigonometric tables to find \(\theta\).

### Final Step: Conclusion
After performing the calculations, we find that the angle between the two lines is:
\[ \theta = 15^\circ \]