I’m working through Adams Calculus a Complete Course ninth edition; on page 19 the parabola with focus point and directrix
is given as
, the vertex is
.
I was wondering what the focus point and directrix of a general parabola would be. So I took pen and paper for an interlude.
First realise that has
as vertex. Writing
in the form of
will help us find its focus point, directrix and vertex, all expressed in
and
.
If you don’t understand the adding of , just think of
, where
.
The focus point is:
The directrix is:
The vertex is .
The distance between the focal point and the vertex is , which is
. Why would you want to know this? Because you might want to know where to put the pan in the solar oven, or the light bulb in a parabola shaped mirror. The funny thing is that a mathematician might be interested in it just because it is possible to calculate this distance, so for no practical reason.
Just in case that you wonder if it is wright that is a parabola with
as vertex. Let us take focus point
and line
, now let us see where the points
are that have the same distance to the line and the focus point: