JEE Challenger
More from Straight Lines
MathMedAdv 2008 • Paper 2

Match Conditions for Three Straight Lines Being Concurrent Parallel or Forming Triangle

Consider the lines given by
L1:x+3y−5=0L_1 : x+3y-5=0
L2:3x−ky−1=0L_2 : 3x-ky-1=0
L3:5x+2y−12=0L_3 : 5x+2y-12=0

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4×44 \times 4 matrix given in the ORS.

Column IColumn II(A) L1,L2,L3 are concurrent, if(p) k=−9(B) One of L1,L2,L3 is parallel to at least one of the other two, if(q) k=−65(C) L1,L2,L3 form a triangle, if(r) k=56(D) L1,L2,L3 do not form a triangle, if(s) k=5\begin{array}{ll} \text{Column I} & \text{Column II} \\ \text{(A) } L_1, L_2, L_3 \text{ are concurrent, if} & \text{(p) } k = -9 \\ \text{(B) One of } L_1, L_2, L_3 \text{ is parallel to at least one of the other two, if} & \text{(q) } k = -\frac{6}{5} \\ \text{(C) } L_1, L_2, L_3 \text{ form a triangle, if} & \text{(r) } k = \frac{5}{6} \\ \text{(D) } L_1, L_2, L_3 \text{ do not form a triangle, if} & \text{(s) } k = 5 \end{array}

Options

A

(A)→(S),  (B)→(P),  (C)→(R),  (D)→(P,Q,S)(\text{A}) \rightarrow (\text{S}),\; (\text{B}) \rightarrow (\text{P}),\; (\text{C}) \rightarrow (\text{R}),\; (\text{D}) \rightarrow (\text{P}, \text{Q}, \text{S})

B

(A)→(P,S),  (B)→(P,Q),  (C)→(R),  (D)→(Q,S)(\text{A}) \rightarrow (\text{P}, \text{S}),\; (\text{B}) \rightarrow (\text{P}, \text{Q}),\; (\text{C}) \rightarrow (\text{R}),\; (\text{D}) \rightarrow (\text{Q}, \text{S})

C

(A)→(S),  (B)→(P,Q),  (C)→(R),  (D)→(P,Q,S)(\text{A}) \rightarrow (\text{S}),\; (\text{B}) \rightarrow (\text{P}, \text{Q}),\; (\text{C}) \rightarrow (\text{R}),\; (\text{D}) \rightarrow (\text{P}, \text{Q}, \text{S})

Correct
D

(A)→(S),  (B)→(Q),  (C)→(R),  (D)→(P,S)(\text{A}) \rightarrow (\text{S}),\; (\text{B}) \rightarrow (\text{Q}),\; (\text{C}) \rightarrow (\text{R}),\; (\text{D}) \rightarrow (\text{P}, \text{S})

Topics & Concepts

Straight LinesStraight LinesFamily of Lines

Step-by-Step Solution

To determine the conditions under which the given lines L1,L2,L3L_1, L_2, L_3 are concurrent, parallel, or form/do not form a triangle, we analyze the lines:
L1:x+3y−5=0L_1 : x + 3y - 5 = 0
L2:3x−ky−1=0L_2 : 3x - ky - 1 = 0
L3:5x+2y−12=0L_3 : 5x + 2y - 12 = 0

1. Finding the Point of Intersection of L1L_1 and L3L_3

Solving the equations of L1L_1 and L3L_3 simultaneously:
From L1L_1, we have:
x=5−3yx = 5 - 3y

Substitute xx into the equation for L3L_3:
5(5−3y)+2y−12=05(5 - 3y) + 2y - 12 = 0
25−15y+2y−12=025 - 15y + 2y - 12 = 0
13−13y=0  ⟹  y=113 - 13y = 0 \implies y = 1

Substitute y=1y = 1 back into x=5−3yx = 5 - 3y:
x=5−3(1)=2x = 5 - 3(1) = 2

Thus, the lines L1L_1 and L3L_3 intersect at the unique point (2,1)(2, 1).

2. Condition for Concurrency (Statement A)

The three lines L1,L2,L3L_1, L_2, L_3 are concurrent if the intersection point (2,1)(2, 1) of L1L_1 and L3L_3 lies on L2L_2:
3(2)−k(1)−1=03(2) - k(1) - 1 = 0
6−k−1=0  ⟹  k=56 - k - 1 = 0 \implies k = 5

Therefore, (A) →\rightarrow (s).

3. Condition for Parallelism (Statement B)

The slopes of the lines are:

  • Slope of L1L_1: m1=−13m_1 = -\frac{1}{3}
  • Slope of L2L_2: m2=3km_2 = \frac{3}{k} (for k≠0k \neq 0)
  • Slope of L3L_3: m3=−52m_3 = -\frac{5}{2}

Two lines are parallel if their slopes are equal:

  • L1∥L2  ⟹  m1=m2  ⟹  −13=3k  ⟹  k=−9L_1 \parallel L_2 \implies m_1 = m_2 \implies -\frac{1}{3} = \frac{3}{k} \implies k = -9
  • L2∥L3  ⟹  m2=m3  ⟹  3k=−52  ⟹  k=−65L_2 \parallel L_3 \implies m_2 = m_3 \implies \frac{3}{k} = -\frac{5}{2} \implies k = -\frac{6}{5}
  • L1∥L3  ⟹  m1=m3  ⟹  −13=−52L_1 \parallel L_3 \implies m_1 = m_3 \implies -\frac{1}{3} = -\frac{5}{2} (not possible)

Therefore, one of the lines is parallel to another if k=−9k = -9 or k=−65k = -\frac{6}{5}.

Thus, (B) →\rightarrow (p), (q).

4. Condition for Forming a Triangle (Statement C)

Three straight lines form a triangle if and only if:

  1. No two lines are parallel to each other.
  2. The three lines are not concurrent.

For k=56k = \frac{5}{6}:

  • Slope of L2L_2: m2=35/6=185≠−13m_2 = \frac{3}{5/6} = \frac{18}{5} \neq -\frac{1}{3} and m2≠−52m_2 \neq -\frac{5}{2}. Hence, no two lines are parallel.
  • k=56≠5k = \frac{5}{6} \neq 5, so the lines are not concurrent.

Thus, the lines form a triangle for k=56k = \frac{5}{6}.

Therefore, (C) →\rightarrow (r).

5. Condition for NOT Forming a Triangle (Statement D)

Three lines fail to form a triangle if at least two lines are parallel or if all three lines are concurrent. From the analyses above, this occurs when:

  • k=−9k = -9 (parallel lines L1∥L2L_1 \parallel L_2)
  • k=−65k = -\frac{6}{5} (parallel lines L2∥L3L_2 \parallel L_3)
  • k=5k = 5 (concurrent lines)

Therefore, (D) →\rightarrow (p), (q), (s).

Final Match:

  • (A) →\rightarrow (s)
  • (B) →\rightarrow (p), (q)
  • (C) →\rightarrow (r)
  • (D) →\rightarrow (p), (q), (s)
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