The angle between the line x = a and `by+c= 0`, is

Options

A. `0^(@)`
B. `60^(@)`
C. `180^(@)`
D. `90^(@)`

Answer

To find the angle between the line \( x = a \) and the line \( by + c = 0 \), we can follow these steps:

### Step 1: Identify the lines
The first line is given as \( x = a \). This line is vertical and is parallel to the y-axis.

The second line is given as \( by + c = 0 \). We can rearrange this equation to find its slope.

### Step 2: Rearranging the second line
Rearranging \( by + c = 0 \) gives:
\[ by = -c \implies y = -\frac{c}{b} \]
This shows that the line is horizontal and is parallel to the x-axis.

### Step 3: Determine the slopes
- The slope of the line \( x = a \) (vertical line) is undefined or can be considered as infinite.
- The slope of the line \( y = -\frac{c}{b} \) (horizontal line) is 0.

### Step 4: Use the angle formula
The angle \( \theta \) between two lines can be calculated using the formula:
\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
where \( m_1 \) and \( m_2 \) are the slopes of the two lines.

In our case:
- \( m_1 \) (slope of the vertical line) is undefined (or infinite).
- \( m_2 \) (slope of the horizontal line) is 0.

### Step 5: Analyze the angle
Since one line is vertical and the other is horizontal, the angle between them is \( 90^\circ \).

### Conclusion
Thus, the angle between the line \( x = a \) and the line \( by + c = 0 \) is:
\[ \text{Angle} = 90^\circ \]

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