The angle between the two lines `y-2x=9 and x+2y =-7,` is

Options

A. `60^(@)`
B. `30^(@)`
C. `90^(@)`
D. `45^(@)`

Answer

To find the angle between the two lines given by the equations \(y - 2x = 9\) and \(x + 2y = -7\), we can follow these steps:

### Step 1: Convert the equations to slope-intercept form
We start by rewriting each equation in the form \(y = mx + b\), where \(m\) is the slope.

1. For the first line \(y - 2x = 9\):
\[ y = 2x + 9 \]
Here, the slope \(m_1 = 2\).

2. For the second line \(x + 2y = -7\):
\[ 2y = -x - 7 \implies y = -\frac{1}{2}x - \frac{7}{2} \]
Here, the slope \(m_2 = -\frac{1}{2}\).

### Step 2: Use the formula for the angle between two lines
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
Substituting \(m_1 = 2\) and \(m_2 = -\frac{1}{2}\) into the formula:
\[ \tan \theta = \left| \frac{2 - (-\frac{1}{2})}{1 + 2 \cdot (-\frac{1}{2})} \right| \]
This simplifies to:
\[ \tan \theta = \left| \frac{2 + \frac{1}{2}}{1 - 1} \right| = \left| \frac{\frac{4}{2} + \frac{1}{2}}{0} \right| = \left| \frac{\frac{5}{2}}{0} \right| \]

### Step 4: Analyze the result
Since the denominator is zero, \(\tan \theta\) approaches infinity, which indicates that the lines are perpendicular to each other.

### Conclusion
The angle between the two lines is \(90^\circ\).