Two parabolas in a plane are given, such that they don't intersect. Is it true that there is a line in plane such that doesn't intersect any of them?
If two parabolas don't intersect, is there a line that doesn't intersect either of them?
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algebra-precalculus
conic-sections
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0Do you mean parabolas as graphs of functions $y=f(x) = ax^2 + bx + c$? Or are we talking about general parabolas? – 2012-06-07
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0Several errors still remain at this time in the answer by Robert Mastragostino. – 2012-06-07
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0From a certain point of view, a parabola is an ellipse that has the line at infinity as one of its tangent lines. For the parabola $y=x^2$, if we pick a different line to serve as the line at infinity, namely $y=-1$, then the parabola becomes an ellipse. So if two parabolas don't intersect, the question is whether there is some line we could choose to regard as being at infinity, such that _both_ parabolas become ellipses. If two ellipses in the projective plane don't intersect, we can always find a line far far away from both of them. – 2012-06-07