The angle between the line joining the points (1,-2) , (3,2) and the line x+2y -7 =0 is

Options

A. `pi`
B. `pi//2`
C. `pi//3`
D. `pi//6`

Explanation

To find the angle between the line joining the points (1, -2) and (3, 2) and the line given by the equation \(x + 2y - 7 = 0\), we can follow these steps:

### Step 1: Find the slope of the line joining the points (1, -2) and (3, 2).

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (1, -2)\) and \((x_2, y_2) = (3, 2)\).

Calculating the slope:

\[ m_1 = \frac{2 - (-2)}{3 - 1} = \frac{2 + 2}{3 - 1} = \frac{4}{2} = 2 \]

### Step 2: Find the slope of the line given by the equation \(x + 2y - 7 = 0\).

To find the slope from the equation \(x + 2y - 7 = 0\), we can rewrite it in slope-intercept form \(y = mx + b\):

\[ 2y = -x + 7 \quad \Rightarrow \quad y = -\frac{1}{2}x + \frac{7}{2} \]

Thus, the slope \(m_2\) of this line is:

\[ m_2 = -\frac{1}{2} \]

### Step 3: Use the formula for the angle between two lines.

The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) can be found using the formula:

\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

Substituting the values of \(m_1\) and \(m_2\):

\[ \tan \theta = \left| \frac{2 - \left(-\frac{1}{2}\right)}{1 + 2 \cdot \left(-\frac{1}{2}\right)} \right| = \left| \frac{2 + \frac{1}{2}}{1 - 1} \right| \]

Calculating the numerator:

\[ 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]

Calculating the denominator:

\[ 1 - 1 = 0 \]

Since the denominator is zero, this indicates that the lines are perpendicular to each other.

### Conclusion:

The angle between the line joining the points (1, -2) and (3, 2) and the line \(x + 2y - 7 = 0\) is \(90^\circ\).

---