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The sides of a cyclic quadrilateral lie along the lines $$a_1 x + b_1 y = 2013$$, $$a_2 x + b_2 y = 2014$$, $$a_1 x + b_1 y = 2015$$, $$a_2 x + b_2 y = 2018$$ then $$a_1 a_2 + b_1 b_2 =$$ _____________
Correct Answer: 0
The first and third lines are parallel, and so are the second and fourth, so the quadrilateral is a parallelogram. A cyclic parallelogram is a rectangle, so the two families of lines are perpendicular. Lines with normals $$(a_1, b_1)$$ and $$(a_2, b_2)$$ are perpendicular when $$a_1 a_2 + b_1 b_2 = 0$$.
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