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Q.

Three families of lines are as follows 
(i)  2x+3y+1+k1(2x+4y+4)=0,       (ii)  (3+6tanθ)x+(4+4tanθ)y+4+tanθ=0,      (iii)  (x+y+λ)+k3(2x+y+1)=0 
Then the values of λ so that 3 families have a common member is  pq, where p and q are positive integers and  co-prime to each other, then the value of (4q3p) is

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answer is 7.

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Detailed Solution

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For first family the fixed point is (4,3).
For second family fixed point is (1,74) and third family has fixed point as  (λ1,12λ)
Now for three families to have a common member, these three points should be collinear so 124311741λ112λ1=0

4(74(12λ)+3(1λ+1))+((12λ)(74)(λ1))=0

This gives  λ=2319

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