Question

Let A(1,3) and B(5,1) be two points. If a line with slope m intersects AB at an angle of 45\(^\circ\), then the possible values of m are

• A. \( 5, \frac{-1}{5} \)
• B. \( 7, \frac{1}{7} \)
• C. \( 3, \frac{1}{3} \)
• D. \( -3, \frac{1}{3} \)

Hint:

When dealing with the formula for the angle between two lines, the absolute value is crucial. Remember that it leads to two separate linear equations, yielding two possible slopes for the second line (unless the lines are parallel or perpendicular).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves finding the slope of a line given the angle it makes with another line. The key is to use the formula for the angle between two lines, which relates their slopes.
Step 2: Key Formula or Approach:
The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by the formula: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \] Step 3: Detailed Explanation:
First, we need to find the slope of the line segment AB. Let's call this slope \( m_{AB} \). Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), with points A(1,3) and B(5,1): \[ m_{AB} = \frac{1 - 3}{5 - 1} = \frac{-2}{4} = -\frac{1}{2} \] Let the slope of the other line be \( m \). We are given that the angle \( \theta \) between the lines is 45\(^\circ\). We know that \( \tan(45^\circ) = 1 \). Substituting the values into the formula: \[ 1 = \left| \frac{m - m_{AB}}{1 + m \cdot m_{AB}} \right| = \left| \frac{m - (-\frac{1}{2})}{1 + m(-\frac{1}{2})} \right| = \left| \frac{m + \frac{1}{2}}{1 - \frac{m}{2}} \right| \] To simplify, multiply the numerator and denominator inside the absolute value by 2: \[ 1 = \left| \frac{2m + 1}{2 - m} \right| \] This equation gives two possibilities: Case 1: \( \frac{2m + 1}{2 - m} = 1 \) \[ 2m + 1 = 2 - m \] \[ 3m = 1 \] \[ m = \frac{1}{3} \] Case 2: \( \frac{2m + 1}{2 - m} = -1 \) \[ 2m + 1 = -(2 - m) = -2 + m \] \[ m = -3 \] Step 4: Final Answer:
The possible values of m are \( \frac{1}{3} \) and -3.