Question

The obtuse angle between lines \(2y = x + 1\) and \(y = 3x + 2\) is:

• A. \(3\pi/4\)
• B. \(5\pi/6\)
• C. \(4\pi/3\)
• D. \(2\pi/3\)

Answer 1

The Correct Option is A

Step 1: Write both equations in slope-intercept form \(y = mx + c\). 
For \(2y = x + 1 \Rightarrow y = \frac{1}{2}x + \frac{1}{2}\). Hence, slope \(m_1 = \frac{1}{2}\).
For \(y = 3x + 2\), slope \(m_2 = 3\). Step 2: The acute angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by: \[ \tan\theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \] Substitute values: \[ \tan\theta = \left| \frac{3 - \frac{1}{2}}{1 + 3 \times \frac{1}{2}} \right| = \left| \frac{\frac{5}{2}}{\frac{5}{2}} \right| = 1 \] \[ \Rightarrow \theta = \frac{\pi}{4} \] Step 3: The question asks for the obtuse angle between the lines. The obtuse angle is: \[ \pi - \frac{\pi}{4} = \frac{3\pi}{4} \] Final Answer: \[ \boxed{\text{Obtuse angle} = \frac{3\pi}{4}} \] Correct Option: (a)