Question

The slopes of the lines, which make an angle 45° with the line \( 3x - y = -5 \), are:

• A. \( 1, -1 \)
• B. \( \frac{1}{2}, -1 \)
• C. \( 1, \frac{1}{2} \)
• D. \( -2, \frac{1}{2} \)

Hint:

To find the slopes of lines making a given angle with another line, use the angle formula and solve for the slopes.

Answer 1

The Correct Option is D

We are given the line \( 3x - y = -5 \), and we are asked to find the slopes of the lines that make an angle of 45° with this line. Step 1: Find the slope of the given line The slope of the given line \( 3x - y = -5 \) is: \[ \text{slope} = \frac{3}{1} = 3 \] Step 2: Use the formula for the angle between two lines The formula for the angle between two lines with slopes \( m_1 \) and \( m_2 \) is: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \] Since the angle is 45°, we have \( \tan 45^\circ = 1 \): \[ 1 = \left| \frac{3 - m}{1 + 3m} \right| \] Solve for \( m \): \[ \frac{3 - m}{1 + 3m} = 1 \quad \Rightarrow \quad 3 - m = 1 + 3m \] \[ 2 = 4m \quad \Rightarrow \quad m = \frac{1}{2} \] Similarly, for the other line making an angle of 45°, we solve: \[ \frac{3 - m}{1 + 3m} = -1 \quad \Rightarrow \quad 3 - m = -(1 + 3m) \] \[ 4 = 2m \quad \Rightarrow \quad m = -2 \] Thus, the slopes are \( -2 \) and \( \frac{1}{2} \). Thus, the correct answer is \( -2, \frac{1}{2} \).