The angle between the lines y – x + 5 = 0 and ` sqrt(3) x - y + 7 = 0` is

Options

A. `15^@`
B. `60^@`
C. `45^@`
D. `75^@`

Answer

To find the angle between the lines given by the equations \( y - x + 5 = 0 \) and \( \sqrt{3}x - y + 7 = 0 \), we will follow these steps:

### Step 1: Write the equations in slope-intercept form
We need to convert both equations into the form \( y = mx + c \) where \( m \) is the slope.

1. For the first line:
\[ y - x + 5 = 0 \implies y = x - 5 \]
Here, the slope \( m_1 = 1 \).

2. For the second line:
\[ \sqrt{3}x - y + 7 = 0 \implies y = \sqrt{3}x + 7 \]
Here, the slope \( m_2 = \sqrt{3} \).

### Step 2: Use the angle formula
The formula to find 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
Now we substitute \( m_1 = 1 \) and \( m_2 = \sqrt{3} \) into the formula:
\[ \tan \theta = \left| \frac{1 - \sqrt{3}}{1 + 1 \cdot \sqrt{3}} \right| \]
This simplifies to:
\[ \tan \theta = \left| \frac{1 - \sqrt{3}}{1 + \sqrt{3}} \right| \]

### Step 4: Calculate the value
To find \( \tan \theta \):
1. Calculate \( 1 - \sqrt{3} \) and \( 1 + \sqrt{3} \):
- \( 1 - \sqrt{3} \approx 1 - 1.732 = -0.732 \)
- \( 1 + \sqrt{3} \approx 1 + 1.732 = 2.732 \)

2. Thus,
\[ \tan \theta = \left| \frac{-0.732}{2.732} \right| \approx 0.268 \]

### Step 5: Find the angle
Now, we can find \( \theta \) using the inverse tangent function:
\[ \theta = \tan^{-1}(0.268) \]
Using a calculator, we find:
\[ \theta \approx 15^\circ \]

### Final Answer
The angle between the lines is \( 15^\circ \).

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