Question

The slope of aline is double of the slope of another line. If tangent of the angle between hem is $\frac{1}{3}$. Find the slopes of the lines.

• A. -1 and -2
• B. and -1
• C. 1 and 2
• D. and 1

Answer 1

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

The correct options are
A

-1 and -2

B

and -1

C

1 and 2

D

and 1

Let $m_1\&m_2$ be the slopes of the two given lines such that $m_2=mand{m}_1=2m$
$\theta$ is the angle between the lines $l_{1}and{l}_2$ with slope $m_1\&m_2$ then $tan\theta =|\frac{m_1-m_2}{1+m_1.m_2}|$
If is given that the tangent of angle between the two lines is $\frac{1}{3}$
$\frac{1}{3}=\frac{-m}{1+2m^2}\frac{1}{3}=\frac{-(-m)}{1+2m^2}$
$(2m^2+1)=-3m\frac{1}{3}=\frac{m}{1+2m^2}$
$2m^2+3m+11+2m^2=3m$
$(m+1)(2m+1)=02m^2-3m+1=0$
$m=-1,\frac{-1}{2}(m-1)(2m-1)=0$
$m=1,\frac{1}{2}$
If m=-1, then the slopes of the lines are -1 and -2 if m=1, then the slope of lines are 1 and 2
If $m=\frac{-1}{2}$, then the slopes of the lines are $\frac{-1}{2}$ and -1 if $m=\frac{1}{2}$, then the slope of lines are $\frac{1}{2}$ and 1

Hence the slopes of the lines are -1 and -2 or $\frac{-1}{2}$ and -1 or 1 and 2 or $\frac{1}{2}$ and 1.