Question

The slope of lines which makes an angle \( 45^\circ \) with the line \( 2x - y = -7 \)

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

Hint:

Use the angle between two lines formula: \( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \) to find the required slope.

Answer 1

The Correct Option is A

Given line: \( 2x - y = -7 \Rightarrow y = 2x + 7 \)
So, slope \( m_1 = 2 \)
Let \( m_2 \) be the slope of the required line making \( 45^\circ \) with the given line. Then,
\[ \tan(45^\circ) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = 1 \Rightarrow \left| \frac{2 - m_2}{1 + 2m_2} \right| = 1 \] Solving:
Case 1: \( \frac{2 - m_2}{1 + 2m_2} = 1 \Rightarrow 2 - m_2 = 1 + 2m_2 \Rightarrow 1 = 3m_2 \Rightarrow m_2 = \frac{1}{3} \)
Case 2: \( \frac{2 - m_2}{1 + 2m_2} = -1 \Rightarrow 2 - m_2 = -1 - 2m_2 \Rightarrow 3 = -m_2 \Rightarrow m_2 = -3 \)
Hence, required slopes are \( \frac{1}{3}, -3 \)