Define ellipse and hyperbola in terms of distances from a point

Question

Just as we say a circle is a locus of points that are equidistant from a single point.

How to define an ellipse and a hyperbola in a similar way?

Answer 1

Let $d(X, Y)$ be the distance between the points $X$ and $Y$ in the plane. Consider two points $P, Q$ in the plane (the foci), and let $c>0$.

The ellipse is the locus of those $M$ such that $d(M,P) + d(M,Q) = c$.

The hyperbola is the locus of those points $M$ such that $|d(M,P) - d(M,Q)| = c$.

Answer 2

Ellipses and hyperbolas cannot be defined with respect to a single center as you expect. It takes either two centers or a center and axis to define them. In the former case it has to be their sum or difference, and in the latter ratio of distances to the defining point.

Hope you find relation between two segment lengths, as others defined.