Straight Lines and Pair of Straight Lines

We have the lines:
L1 : x + 3y - 5 = 0
L2 : 3x - ky - 1 = 0
L3 : 5x + 2y - 12 = 0
(A)→(S)The point of the intersection of L1 and L3 is (2, 1).
If the lines are concurrent, then (2, 1) satisfy L2. Therefore,
3(2) - (1) -1 = 0⇒ = 5
(B) → (P), (Q)
If L1 and L2 are parallel:

3

1

=

−k

3

≠

−1

−5

⇒ k = - 9
If L2 and L3 are parallel:
Hence,

3

5

= −

k

2

⇒ k = −

6

5

(C)→(R)
Since the lines form a triangle, they cannot be concurrent and no two of them are parallel
⇒ k ≠ 5 , - 9 , −

6

5

⇒ k =

5

6

(D)→(P), (Q), (S)
If the lines do not form a triangle,
k = 5 , −9,−

6

5