What angle does the line segment joining (5 , 2) and (6 , -15) subtend at (0 ,0) ?

Options

A. `(pi)/(6)`
B. `(pi)/(4)`
C. `(pi)/(2)`
D. `(3pi)/(4)`

Answer

To find the angle that the line segment joining the points (5, 2) and (6, -15) subtends at the origin (0, 0), we can follow these steps:

### Step 1: Find the slopes of the lines
First, we need to find the slopes of the lines joining the points (5, 2) and (0, 0), and (6, -15) and (0, 0).

1. **Slope of line joining (5, 2) and (0, 0)**:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{5 - 0} = \frac{2}{5} \]

2. **Slope of line joining (6, -15) and (0, 0)**:
\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-15 - 0}{6 - 0} = \frac{-15}{6} = -\frac{5}{2} \]

### Step 2: 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:
\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the slopes into the formula
Substituting \( m_1 = \frac{2}{5} \) and \( m_2 = -\frac{5}{2} \):
\[ \tan(\theta) = \left| \frac{\frac{2}{5} - \left(-\frac{5}{2}\right)}{1 + \frac{2}{5} \cdot \left(-\frac{5}{2}\right)} \right| \]

### Step 4: Simplify the expression
1. **Numerator**:
\[ \frac{2}{5} + \frac{5}{2} = \frac{2 \cdot 2 + 5 \cdot 5}{5 \cdot 2} = \frac{4 + 25}{10} = \frac{29}{10} \]

2. **Denominator**:
\[ 1 + \frac{2}{5} \cdot \left(-\frac{5}{2}\right) = 1 - 1 = 0 \]

### Step 5: Analyze the result
Since the denominator is 0, this implies that \( \tan(\theta) \) approaches infinity. This situation occurs when \( \theta = \frac{\pi}{2} \) radians.

### Conclusion
Thus, the angle \( \theta \) that the line segment joining (5, 2) and (6, -15) subtends at the origin (0, 0) is:
\[ \theta = \frac{\pi}{2} \]

### Final Answer
The angle is \( \frac{\pi}{2} \) radians.

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