conic sections

The parabola is a plane curve defined as the set of all points that are equidistant from a fixed point \(F \) called the focus, and a fixed line \( d \) called the directrix.

An ellipse is the set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. An ellipse is described by the equation: \[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

Given two fixed points in the plane, \( F_1 \) and \( F_2 \), a hyperbola is defined as the set of all points \( P \) such that the absolute value of the difference between the distances from \( P \) to each focus is constant.