So suppose we have a parabolic curve and we want to find out stuff about it.
Its equation … Oh, we have no axes.
Its focus … That would be nice, but it is a bit out of reach.
Its axis, in fact its axis of symmetry … Fold it in half? But how?
Try the method of part one, with the ellipse. (previous post)
This looks promising. I even get another axis, for my coordinate system, if I really want the equation.
Now, analysis of the standard equation for a parabola (see later) says that a line at 45 deg to the axis, as shown, cuts the parabola at a point four focal lengths from the axis. In the picture, marked on the “vertical”axis, this is the length DH
So I need a point one quarter of the way from D to H. Easy !
and then the circle center D, with radius DH/4 cuts the axis of the parabola at the focal point (the focus).
Even better, we get the directrix as well …
and now for the mathy bit (well, you do the algebra, I did the picture)
Yes, I know that this one points up and the previous one pointed to the right !
All diagrams were created with my geometrical construction program, GEOSTRUCT
You will find it here: