Consider the lines given by $$ L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0 $$ Match the…
Consider the lines given by
$$
L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0
$$
Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.
(A) r, (B) p,s, (C) s, (D) p,s
(A) s, (B) p,q,r, (C) r, (D) p,q
(A) s, (B) p,q, (C) r, (D) p,q,s
(A) r, (B) p,q, (C) s, (D) p,q
Solution
(A) Solving equations $L_1$ and $L_3$,
$
\begin{aligned}
\frac{x}{-36+10} & =\frac{y}{+12-25}=\frac{1}{2-15} \\
x & =2, y=1
\end{aligned}
$
$L_1, L_2, L_3$ are concurrent, if point $(2,1)$ lies on $L_2$.
$
\therefore \quad 6-k-1=0 \Rightarrow k=5
$
(B) Either $L_1$ is parallel to $L_2$, or $L_3$ is parallel to $L_2$, then
$
\begin{aligned}
& \frac{1}{3}=\frac{3}{-k} \text { or } \frac{3}{5}=\frac{-k}{2} \\
& k=-9 \quad \text { or } \quad k=-\frac{6}{5}
\end{aligned}
$
$
\Rightarrow \quad k=-9 \text { or } k=-\frac{6}{5}
$
(C) $L_1, L_2, L_3$ form a triangle, if they are not concurrent, or not parallel.
$
\therefore \quad k \neq 5,-9,-\frac{6}{5} \Rightarrow k=\frac{5}{6}
$
(D) $L_1, L_2, L_3$ do not form a triangle, if
$
k=5,-9,-\frac{6}{5}
$
