Puzzle in the squared notebook

In a squared notebook draw segments (0,10)-(0,0), (0,9)-(1,0), (0,8)-(2,0), ..., (0,0)-(10,0). (These are shown as segments AB in the figure below. Just drag point A to get the segments in red.) Can you determine the contour of the segments by setting up an equation? (We assume that you continue to draw the segments infinitely for all possible (0,t)-(10-t,0) segments where t runs between 0 and 10.)
By using GeoGebra's Envelope command you can find the algebraic equation of the geometric object. Click on "Show envelope" to get the equation. But here three objects are shown, a 5th order polynomial, which is the algebraic product of the equation of the real curve and its reflection to the x-axis, and also the x-axis itself.
  1. Find the 2th order polynomial which gives the real part of the curve in this problem.
  2. Prove that this curve is a parabola and it is symmetrical to the line y=x. (Click on "Show axes of symmetry" for a basic test.) Also prove that the x- and y-axes are tangents of this parabola.