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One of the questions in my book asked me to find the equation of axis of parabola by giving the 4 points through which parabola passes.
The points were: $(1, 2), (2, 1),(3, 4), (4, 3)$. The solution provided explained that since line joining points $(1, 2), (2, 1)$ and another line joining points $(3, 4), (4, 3)$ are parallel, so the line perpendicular to it would be the axis of parabola. But when I tried drawing it, how can a parabola pass through these 4 points since the length of both lines are equal?
Also, I had studied that locus of mid points of parallel chords of parabola is a straight line parallel to the axis of parabola, but how can the solution say that it would be the axis of parabola?

Shekhar Dangi
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    As mentioned in a comment here by Batominovski: 'I just realized that you don't have a non-degenerate parabola here. The only "parabolic equations" that pass through your points are $(x-y-1)(x-y+1)=0$ and $(x+y-3)(x+y-7)=0$. Each of these two equations just give you a union of two parallel lines.' – VTand Dec 25 '22 at 13:55

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