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Q. Let the point P(α,β) be at a unit distance from each of the two lines L1:3x4y+12=0, and L2:8x+6y+11=0. If P lies below L1 and above L2, then 100(α+β) is equal to

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Solution:

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By observing origin and P lies in same region.
L1(0,0)>0;L1(α,β)>03α4β+12>0 1=|3α4β+125|
3α4β+12=5......(1)
Similarly for L2
L2(0,0)>0;L2(α,β)>0
1=|8α+6β+1110|8α+6β+11=10.......(2)
Solving (1) and (2)
α=2325;β=106100
100(α+β)=100(92100+106100)=14