In sub - part (i) to (x) choose the correct option and in sub - part (xi) to (xy), answer the questions as intructed .
The angle between the lines `x-2=0andx+sqrt(3y)-5=0`:

Options

A. `90^(@)`
B. `120^(@)`
C. `60^(@)`
D. `45^(@)`

Answer

To find the angle between the lines given by the equations \( x - 2 = 0 \) and \( x + \sqrt{3}y - 5 = 0 \), we can follow these steps:

### Step 1: Identify the equations of the lines
The first line is given by:
\[ x - 2 = 0 \]
This represents a vertical line at \( x = 2 \).

The second line can be rewritten as:
\[ x + \sqrt{3}y - 5 = 0 \]
This can be rearranged to express \( y \) in terms of \( x \):
\[ \sqrt{3}y = 5 - x \]
\[ y = \frac{5 - x}{\sqrt{3}} \]

### Step 2: Determine the slopes of the lines
For a line in the form \( y = mx + c \), the slope \( m \) can be identified directly.

1. The first line \( x - 2 = 0 \) is vertical, which means its slope \( m_1 \) is undefined (or can be considered as infinity).

2. The second line \( y = \frac{5}{\sqrt{3}} - \frac{1}{\sqrt{3}}x \) has a slope:
\[ m_2 = -\frac{1}{\sqrt{3}} \]

### Step 3: Use the formula for the angle between two lines
The formula for the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

Since \( m_1 \) is undefined (infinity), we can use the fact that the angle between a vertical line and any other line can be directly calculated using the slope of the second line.

### Step 4: Calculate the angle
Since the first line is vertical, the angle \( \theta \) between the vertical line and the second line can be found using:
\[ \tan \theta = \left| m_2 \right| = \left| -\frac{1}{\sqrt{3}} \right| = \frac{1}{\sqrt{3}} \]
This corresponds to:
\[ \theta = 30^\circ \]

### Conclusion
Thus, the angle between the two lines is \( 30^\circ \).

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