I was observing the standard equations of a parabola and an ellipse, and I realised that both equations are very similar but they have different signs. Why is that? How can you demonstrate?
$\endgroup$ 11 Answer
$\begingroup$As someone famous once said, if you change a sign in an equation, there would be no light. There might be some kind of intuition suggesting simple symmetry, which just does not hold when variables are included. Two divergent functions added (in absolute sense) will diverge, but if subtracted, they might converge. Do not get fooled by a "simple" change of a sign.
For parabola and ellipse, lets examine the geometrical definition of booth curves. An ellipse is defined as a set of points, where sum of distance from two points is constant - there comes the sing in the equation: $a+b=const$. A parabola is defined as a set of points, where the distance of the point from a focus and x-axis are the same (in the simple case) - the equation is of form $a-b=0$. It might seem as a simple switch, however the consequences are great.
$\endgroup$ 1
