Equations of conic sections

Here we will have a look at three different conic sections:


1. Parabola

The parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. The equation for a parabola is

$$y=a(x-b)^{2}+c\; or\; x=a(y-b)^{2}+c$$

2. Circles and ellipses

The equation of a circle with center at (a,b) and radius r units is


An ellipse is the figure consisting of all points in the plane whose coordinates satisfy the equation


If the ellipse has its center at (m,n) the equation could be written as


3. Hyperbolas

A hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows.

The equation of a hyperbola with a center at (m,n) is


Video lesson

Write the given equation on the standard equation form for ellipses