Question

If $L_1$ and $L_2$ be two non-vertical lines with slopes $m_1$ and $m_2$ respectively. If ${Î\pm }_1$ and ${Î\pm }_2$ are the inclinations of lines $L_1$ and $L_2$, respectively. Let $θ$ and $Ï•$ be the adjacent angles between the lines $L_1$ and $L_2$. Then, match the terms of Column I with terms of Column II and choose the correct option from the codes given below.

Column I		Column II
A	$\tan θ$	1	$\frac{m_1−m_2}{1+m_1{m}_2}($ as $1+m_1{m}_2î€=0)$
B	$\tan Ï•$	2	$\frac{m_2−m_1}{1+m_1{m}_2}$ is positive,(as $1+m_1{m}_2î€=0$ )
C	$θ$ will be acute and $Ï•$ will be obtuse	3	$\frac{m_2−m_1}{1+m_1{m}_2}$ is negative, (as $1+m_1{m}_2î€=0$ )
D	$θ$ will be obtuse and $Ï•$ will be acute	4	$\frac{m_2−m_1}{1+m_1{m}_2}($ as $1+m_1{m}_2î€=0)$

• A. A - (1), B - (4), C -(2), D -(3)
• B. A - (4), B - (1), C -(2), D -(3)
• C. A - (4), B - (1), C -(3), D -(2)
• D. A - (4), B - (2), C -(1), D -(3)

Answer 1

Answer: (B)

Let $L_1$ and $L_2$ be two non-vertical lines with slopes $m_1$ and $m_2$, respectively. If ${Î\pm }_1$ and ${Î\pm }_2$ are the inclinations of lines $L_1$ and $L_2$ respectively, then
$m_1=\tan {Î\pm }_1\text{ and }{m}_2=\tan {Î\pm }_2$
Let $θ$ and $ϕ$ be the adjacent angles between the lines $L_1$ and $L_2$. Then,
$θ={Î\pm }_2−{Î\pm }_1$ and ${Î\pm }_1,{Î\pm }_2î€=90^{∘}$ (exterior angle theorem)
A. Therefore, $\tan θ=\tan \left({Î\pm }_2−{Î\pm }_1\right)$
$=\frac{\tan {Î\pm }_2−\tan {Î\pm }_1}{1+\tan {Î\pm }_1\tan {Î\pm }_2}=\frac{m_2−m_1}{1+m_1{m}_2},\text{ as }1+m_1{m}_2î€=0$
B. Since, $ϕ=180^{∘}−θ$
$∴\tan ϕ=\tan \left(180^{∘}−θ\right)=−\tan θ$
$=−\frac{m_2−m_1}{1+m_1{m}_2},\text{ as }1+m_1{m}_2î€=0$

Now, here arise two cases.
C. Case I If $\frac{m_2−m_1}{1+m_1{m}_2}$ is positive, then $\tan θ$ will be positive and tan $ϕ$ will be negative, which means $θ$ will be acute and $ϕ$ will be obtuse.
D. Case II If $\frac{m_2−m_1}{1+m_1{m}_2}$ is negative, then $\tan θ$ will be negative and $\tan ϕ$ will be positive, which means that $θ$ will be obtuse and $ϕ$ will be acute.