If the sum of all values of ‘K’ for which the linesL1:x+y+3=0 , L2:2x+ky−1=0 and L3:2x+3y+1=0 do not form a triangle is mn,(m,n∈N). Then value of mn is, where [K] denotes greatest integer less than or equal to K.

ConcurrentK=175, L1∥L2⇒k=2, L2∥L3⇒k=3 ∴mn=175+2+3=425; mn=8

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If the sum of all values of ‘K’ for which the lines $L_{1} : x + y + 3 = 0$ , $L_{2} : 2 x + k y - 1 = 0$ and $L_{3} : 2 x + 3 y + 1 = 0$ do not form a triangle is $\frac{m}{n}, (m, n \in N)$ . Then value of $[\frac{m}{n}]$ is, where [K] denotes greatest integer less than or equal to K.

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answer is 8.

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Concurrent $K = \frac{17}{5}, L_{1} ∥ L_{2} \Rightarrow k = 2, L_{2} ∥ L_{3} \Rightarrow k = 3$

$∴ \frac{m}{n} = \frac{17}{5} + 2 + 3 = \frac{42}{5}; [\frac{m}{n}] = 8$

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