The angle between the straight lines x -y `sqrt3=5 and sqrt3x +y =7 ` is

Options

A. `90^(@)`
B. `60^(@)`
C. `75^(@)`
D. `30^(@)`

Explanation

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

### Step 1: Rewrite the equations in slope-intercept form
We start with the first equation:
\[ x - y\sqrt{3} = 5 \]
Rearranging it gives:
\[ y\sqrt{3} = x - 5 \]
Dividing by \(\sqrt{3}\):
\[ y = \frac{1}{\sqrt{3}}x - \frac{5}{\sqrt{3}} \]
This shows that the slope \( m_1 \) of the first line is:
\[ m_1 = \frac{1}{\sqrt{3}} \]

Now for the second equation:
\[ \sqrt{3}x + y = 7 \]
Rearranging it gives:
\[ y = -\sqrt{3}x + 7 \]
This shows that the slope \( m_2 \) of the second line is:
\[ m_2 = -\sqrt{3} \]

### Step 2: Calculate the product of the slopes
Next, we calculate the product of the slopes:
\[ m_1 \cdot m_2 = \left(\frac{1}{\sqrt{3}}\right) \cdot (-\sqrt{3}) = -1 \]

### Step 3: Determine the angle between the lines
The condition \( m_1 \cdot m_2 = -1 \) indicates that the two lines are perpendicular. Therefore, the angle \( \theta \) between the two lines is:
\[ \theta = 90^\circ \]

### Final Answer
Thus, the angle between the two straight lines is:
\[ \theta = 90^\circ \]

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