Question

If the angle between two lines is \(\frac\pi4\) and slope of one of the lines is \(\frac12\), then the slope of the other line is:

• A. \(-2\)
• B. \(-\frac12\)
• C. 1
• D. 3

Hint:

Always check both positive and negative cases when using the formula for angle between lines, but choose the one matching the given angle’s orientation.

Answer 1

The Correct Option is D

Let slopes of the two lines be \(m_1 = \frac12\) and \(m_2\).
The formula for the angle \(\theta\) between two lines is:
\[ \tan \theta = \left| \fracm_2 - m_11 + m_1 m_2 \right| \] Here, \(\theta = \frac\pi4\), so \(\tan \theta = 1\).
Thus:
\[ 1 = \left| \fracm_2 - \frac121 + \frac12 m_2 \right| \] Case 1: Positive ratio:
\[ \fracm_2 - \frac121 + \frac12 m_2 = 1 \] Multiply through by the denominator:
\(m_2 - \frac12 = 1 + \frac12 m_2\)
\(m_2 - \frac12 m_2 = 1 + \frac12\)
\(\frac12 m_2 = \frac32 \Rightarrow m_2 = 3\)
Case 2: Negative ratio leads to \(m_2 = -\frac13\), which would give a different angle.
Thus, the slope of the other line is \(3\).