This is going to involve a lot of TeX, so will take rather more time thanOriginally Posted by braddy
I have available to do it in one go, so I will do this incrementally.
(PH you were wise to leave your post to be checked, you seem to have
gotten the wrong end of the stick, as we say over here).
You want to find a family of functions/curves which are orthogonal
at every point to the parabola from the given family passing through
Lets start by deriving the equations of the family of parabolas with vertex
at and axes parallel to the vertical axis. Well they
where parameterises the family. Now if one of these
parabolas passess through the point , we have:
and so the parabola through is:
We are interested in the slope of this parabola at as
we are interested in the slope of the curve in the orthoganal family through
the point, which as it is orthogonal to the parabola has a slope of minus the
reciprical of the slope of the parabola at this point.
For the parabola we have:
Hence the slope of the member of the orthogonal family through is:
so the ODE satisfied by the orthogonal trajectory is:
Which is of variables seperable type, and I think you will find that the
solution is a circle with centre .