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 and indicate your answer by darkening the ap

Question

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 and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS. [2008]

	Column I		Column II
(A)	$L_{1}, L_{2}, L_{3}$ are concurrent, if	(p)	$k = - 9$
(B)	One of $L_{1}, L_{2}, L_{3}$ is parallel to at least one of the other two, if	(q)	$k = - \frac{6}{5}$
(C)	$L_{1}, L_{2}, L_{3}$ form a triangle, if	(r)	$k = \frac{5}{6}$
(D)	$L_{1}, L_{2}, L_{3}$ do not form a triangle, if	(s)	$k = 5$

Smart Achievers · Accepted Answer

(1)

(A) → (s); (B) → (p, q); (C) → (r); (D) → (p, q, s)

The given lines are

$L_{1} : x + 3 y - 5 = 0$

$L_{2} : 3 x - k y - 1 = 0$

$L_{3} : 5 x + 2 y - 12 = 0$

(A) Three lines $L_{1}, L_{2}, L_{3}$ are concurrent, if

$| \begin{matrix} 1 & 3 & 5 \\ 3 & - k & 1 \\ 5 & 2 & 12 \end{matrix} |$ $k = - 9$ $\Rightarrow 13 k - 65 = 0$ $\Rightarrow k = 5$

$∴$ $(A) \to (s)$

(B) For $L_{1} ∥ L_{2}$ $\Rightarrow \frac{1}{3} = \frac{- 3}{k}$ $\Rightarrow k = - 9$

and $L_{2} ∥ L_{3}$ $\Rightarrow \frac{3}{5} = \frac{- k}{2}$ $\Rightarrow k = - \frac{6}{5}$

$∴$ $(B) \to (p), (q)$

(C) Three lines $L_{1}, L_{2}, L_{3}$ will form a triangle if no two of them are parallel and no three are concurrent.

$∴$ $k \neq 5, - 9, - \frac{6}{5}$

$∴$ $(C) \to (r)$

(D) Three lines $L_{1}, L_{2}, L_{3}$ do not form a triangle if either any two of these are parallel or the three are concurrent, i.e.,

$k = 5, - 9, - \frac{6}{5}$

$∴$ $(D) \to (p), (q), (s)$

Solution

1 (A) → (s); (B) → (p, q); (C) → (r); (D) → (p, q, s)

2 (A) → (p, q, s); (B) → (p, q); (C) → (r); (D) → (s)

3 (A) → (p, q, s); (B) → (r); (C) → (p, q); (D) → (s)

4 (A) → (r); (B) → (p, q, s); (C) → (p, q); (D) → (s)

Want Us to Email you About Special Offers & Updates?