If y=x2, then (0,0) is the centre.
If 4a(y-k)=(x-h)2, then (h,k) is the centre, where a is the focal length.
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true for apex
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In general, a linear equation CANNOT be made to go through three points. That will only happen if the three points are collinear and in that case, the equation of the line will only require two points.