Given that a + b + c = 0, the line 3ax + by + 2c = 0 always passes through which fixed point?

(2, 2/3)
(2/3, 2)
(-2, 2/3)
(2, 3)

Correct answer: (2, 2/3)

Solution

Substituting c=-(a+b) gives 3ax + by - 2a - 2b = 0, i.e., a(3x-2) + b(y-2) = 0. For this to hold for all a,b: 3x-2=0 and y-2=0, giving x=2/3 and y=2. Wait — that gives (2/3, 2). Let me recheck: 3ax + by + 2c = 0 with c=-a-b: 3ax + by - 2a - 2b = 0 => a(3x-2) + b(y-2) = 0. For all a,b: x=2/3, y=2, i.e., (2/3, 2).

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