Understanding Conic Sections – Parabolas, Circles, Ellipses, and Hyperbolas

Introduction
The following will serve as a brief overview of conic sections or in other words, the functions and graphs associated with the parabola, the circle, the ellipse, and the hyperbola. Initially, it should be noted that these functions are named conic sections since they represent the various ways in which a plane can intersect with a pair of cones.
The Parabola
The first conic section usually studied is the parabola. The equation of a parabola with a vertex at (h, k) and a vertical axis of symmetry is defined as (x – h)^2 = 4p(y – k). Note that if p is positive, the parabola opens upward and if p is negative, it opens downward. For this type of parabola, the focus is centered at the point (h, k + p) and the directrix is a line found at y = k – p.
On the other hand, the equation of a parabola with a vertex at (h, k) and a horizontal axis of symmetry is defined as (y – k)^2 = 4p(x – h). Note that if p is positive, the parabola opens to the right and if p is negative, it opens to the left. For this type of parabola, the focus is centered at the point (h + p, k) and the directrix is a line found at x = h – p.
The Circle
The next conic section to be analyzed is the circle. The equation of a circle of radius r centered at the point (h, k) is given by (x – h)^2 + (y – k)^2 = r^2.
The Ellipse
The standard equation of an ellipse centered at (h, k) is given by [(x – h)^2/a^2] + [(y – k)^2/b^2] = 1 when the major axis is horizontal. In this case, the foci are given by (h +/- c, k) and the vertices are given by (h +/- a, k).
On the other hand, an ellipse centered at (h, k) is given by [(x – h)^2 / (b^2)] + [(y – k)^2 / (a^2)] = 1 when the major axis is vertical. Here, the foci are given by (h, k +/- c) and the vertices are given by (h, k+/- a).
Note that in both types of standard equations for the ellipse, a > b > 0. Also, c^2 = a^2 – b^2. It is important to note that 2a always represents the length of the major axis and 2b always represents the length of the minor axis.
The Hyperbola
The hyperbola is probably the most difficult conic section to draw and understand. By memorizing the following equations and practicing by sketching graphs, one can master even the most difficult hyperbola problem.
To start, the standard equation of a hyperbola with center (h, k) and a horizontal transverse axis is given by [(x – h)^2/a^2] – [(y – k)^2/b^2] = 1. Note that the terms of this equation are separated by a minus sign instead of a plus sign with the ellipse. Here, the foci are given by the points (h +/- c, k), thevertices are given by the points (h +/- a, k) and the asymptotes are represented by y = +/- (b/a)(x – h) +k.
Next, the standard equation of a hyperbola with center (h, k) and a vertical transverse axis is given by [(y- k)^2/a^2] – [(x – h)^2/b^2] = 1. Note that the terms of this equation are separated by a minus sign instead of a plus sign with the ellipse. Here, the foci are given by the points (h, k +/- c), the vertices are given by the points (h, k +/- a) and the asymptotes are represented by y = +/- (a/b)(x – h) + k.
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