Question

The slope of a line is twice the slope of the other line. If the tangent of the angle between them is $\frac{1}{3}$, then the slope of the lines are

• A. $-1$ and $-2$
• B. $-\frac{1}{2}$ and $-1$
• C. $1$ and $2$
• D. $\frac{1}{2}$ and $1$

Answer 1

Answer: (A), (B), (C), (D)

$\frac{1}{2}$ and $1$

Let $m_1,m$ be the slopes of the two given lines such that $m_1=2m$
Let $\theta$ be the angle between them, so
$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$
According to given conditions, we get
$\frac{1}{3}=\left|\frac{m-2m}{1+\left(2m\right)\cdot m}\right|\Rightarrow \frac{1}{3}=\left|\frac{-m}{1+2m^2}\right|\Rightarrow \frac{m}{1+2m^2}=\pm \frac{1}{3}$

Case $1:$ $\frac{m}{1+2m^2}=-\frac{1}{3}$
$\Rightarrow 2m^2+3m+1=0\Rightarrow (m+1)(2m+1)=0\Rightarrow m=-1\text{or}m=-\frac{1}{2}\Rightarrow m_1=-2\text{or}{m}_1=-1$

Case $2:$ $\frac{m}{1+2m^2}=\frac{1}{3}$
$\Rightarrow 2m^2-3m+1=0\Rightarrow (m-1)(2m-1)=0\Rightarrow m=1\text{or}m=\frac{1}{2}\Rightarrow m_1=2\text{or}{m}_1=1$