Question

If the angle between two lines is $\frac{\pi }{3}$ and the slope of one of the lines is $\frac{1}{2}$,
then the slope of the other line is/are:

• A. $8-5\sqrt{3}$
• B. $8+5\sqrt{3}$
• C. $5+8\sqrt{3}$
• D. $5-8\sqrt{3}$

Answer 1

Answer: (A), (B)

$8+5\sqrt{3}$

$\mathrm{Let}{m}_1=\frac{1}{2}\mathrm{and}\theta =\frac{\pi }{3}\mathrm{We}\mathrm{know}\mathrm{that},\tan \theta =\left(\frac{m_2-m_1}{1+m_1{m}_2}\right|\Rightarrow \tan \frac{\pi }{3}=\left(\frac{m_2-\frac{1}{2}}{1+\frac{m_2}{2}}\right|\Rightarrow \sqrt{3}=\left(\frac{2m_2-1}{2+m_2}\right|\Rightarrow \frac{2m_2-1}{2+m_2}=+\sqrt{3}\mathrm{or}\frac{2m_2-1}{2+m_2}=-\sqrt{3}\Rightarrow 2m_2-1=2\sqrt{3}+\sqrt{3}{m}_2\mathrm{or}2m_2-1=-2\sqrt{3}-\sqrt{3}{m}_2\Rightarrow \left(2-\sqrt{3}\right)m_2=2\sqrt{3}+1\mathrm{or}\left(2+\sqrt{3}\right)m_2=1-2\sqrt{3}\Rightarrow m_2=\frac{2\sqrt{3}+1}{2-\sqrt{3}}\mathrm{or}{m}_2=\frac{1-2\sqrt{3}}{2+\sqrt{3}}\Rightarrow m_2=8+5\sqrt{3}\mathrm{or}{m}_2=8-5\sqrt{3}$