What is a hyperbola?

Hyperbola is a collection of all points in a plane, where the difference of the distance from two fixed points is the foci is a constant.

Just as the ellipse, the hyperbola takes the difference between two fixed points instead of the sum compared to that of a ellipse. Using the distance formula to derive at

$$\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ b^{ 2 } } =1\quad and\quad { b }^{ 2 }={ c }^{ 2 }-{ a }^{ 2 }$$