In this note I'll prove that the distance of the focus from the vertex of a parabola has the equation where the parabola has the vertex (h,k).
I would also love feed back on this note. If you look at most of my other notes, I go very far into detail about each step I do to avoid any confusion and to make the note understandable to as many people as possible. However, for the more skilled reader, it may seem a bit long, so this note I'll skip a few simple explanations (no details will be skipped). Tell me what you think in the comments below :) thanks!
Begin with the standard equation of a parabola .
Finding the vertex coordinates using
Our vertex has coordinates
A parabola is the set of points equidistant from the focus and directrix. Thus there is a point (call it m) distance 2p from the focus and distance 2p from the directrix. Therefore, if we draw a line parallel to the directrix from the focus, it will perpendicularly bisect the line from the directrix to m. (Can someone explain why, I can't explain this for some reason, I know why, but I can't put it into words). This is what is depicted in the picture above.
Now, this means there exists a point on our graph distance 2p to the right of our vertex and distance p above. Thus the point has the coordinates . Plugging in for x and y
Bash bash bash
Going back to our first equation
This part is a little tricky so I'll show all the steps. Add to both sides.
Resubstituting EVERYWHERE BUT THE FIRST P.
And we are done.
Remember, please leave feed back on how you liked/disliked the writing style/comprehensiveness of this note.