How Does One Find the Focus of a Parabola?

The focus of a parabola from a given algebraic equation has a distance p from the curve's vertex. To determine the location of the focus, the algebraic equation must be arranged in the form (x-h)^2 = 4p (y-k). From this form, one may extract the value of p and determine the coordinates of the focus.

Before determining the coordinates of the focus, the vertex of the parabola must be identified. From the given algebraic equation of a parabola, (x-h)^2 = 4p (y-k), the vertex is located in the coordinates (h,k). The equation given above applies to a vertical parabola, so the measurement of the distance from the vertex to the focus should also be vertical. Therefore, the set of coordinates for the focus is (h,k+p). In a similar fashion, the focus of a horizontal parabola with the equation (y-k)^2 = 4p (x-h) is located at (h+p,k).

The focus is central to the definition of a class of algebraic curves known as conic sections, which are curves generated from the intersection of a right circular cone with a plane. A parabola consists of a series of points on the intersecting plane that have the same distance from a given point (called the focus) and a given line (called the directrix).