INTRODUCTION TO CONIC SECTIONS
What are conic sections?
Conic sections are geometric figures that are obtained by intersecting a double-napped right circular cone with
a plane. This includes hyperbolas, parabolas, circles, and ellipses.
Identifying Equations of Conic Sections
When given identifying conic sections from their equations, the idea is similar to that of identifying a function
based on the understanding of the equation and graph of the parent function. Equations of circles, ellipses,
hyperbolas, and parabolas have characteristics which allow us to identify the conic based on the equation.
Type of Conic Section
Circle
Parabola
Ellipse
Hyperbola
Equation
(x –
+ (y –
= r2
4p(y – k) = (x – h)2 or 4p(x – h) = (y – k)2
h)2
(𝑥−ℎ)2
𝑎2
(𝑥−ℎ)2
𝑎2
k)2
+
–
(𝑦−𝑘)2
𝑏2
(𝑦−𝑘)2
𝑏2
or
(𝑦−𝑘)2
𝑎2
–
(𝑥−ℎ)2
𝑏2
Circles
A circle is the set of all points (x,y) in a plane that are equidistant from a fixed
point, called the center (focus) of the circle. The distance r between the center
and any point (x, y) on the circle is the radius. The standard equation for a
circle is (x – h)2 + (y – k)2 = r2 where (h, k) is the center and r is the radius.
Parabolas
Up to this point, we have studied parabolas in the context of quadratic
functions. We know parabolas as “U” shaped graphs that either open up or
down. However, there is a more general definition of parabola in regards to conic sections. This general
definition will allow for parabolas that open left and right as well as up and down.
To form a parabola according to ancient Greek definitions, you would start with a point, called the focus, and a
line, called the directrix. The directrix could be a horizontal or vertical line. The parabola is created by all
points (x,y) that form a curve in which each point on that curve is equidistant from the focus and the directrix.
The focus always lies inside the curve and the directrix is always outside of the curve. The vertex of the
parabola is halfway between the focus and the directrix. The axis of symmetry is either a vertical or horizontal
line that cuts the parabola in half crossing both the focus and the vertex. The axis of symmetry is
perpendicular to the directrix.
Standard Equation of a Parabola (Vertex at Origin)
The standard form of the equation of a parabola with vertex at (0,0) and directrix y = –p is
x2 = 4py, p≠0 (“regular” or horizontal parabola)
For directrix x = –p, the equation is
y2 = 4px, p≠0 (“sideways” or vertical parabola)
A more general form of the equation of a parabola is below were (h, k) is the vertex.
regular: 4p(y – k) = (x – h)2
sideways: 4p(x – h) = (y – k)2
Ellipses
An ellipse is the set of all points P in a plane such that the sum of the distances between P and two fixed
points, called the foci, is constant. The line through the foci intersects the ellipse at two vertices. The major
axis joins the vertices. Its midpoint is the ellipse’s center. The line perpendicular to the major axis at the
center intersects the ellipse at the two co-vertices, which are joined by the minor axis. Ellipses can either be
horizontal (a) or vertical (b).
The standard equation for an ellipse is
(𝑥−ℎ)2
𝑎2
+
(𝑦−𝑘)2
𝑏2
where (h, k) is the center of the ellipse, a is the units to
move in the x-direction from the center and b is the number of units to move in the y-direction from the
center. The foci can be found using: a2 – b2 = f2 or denominator – denominator = foci2.
Hyperbolas
A hyperbola is the set of all points P such that the difference of the distances between P and the foci is a
constant. The line through the foci intersects the hyperbola at the two vertices. The transverse axis joins the
two vertices. Its midpoint is the hyperbola’s center. A hyperbola has two branches, and has two asymptotes
that contain the diagonals of a rectangle centered at the hyperbolas center. The standard equation for a
hyperbola is
(𝑥−ℎ)2
𝑎2
–
(𝑦−𝑘)2
𝑏2
or
(𝑦−𝑘)2
𝑎2
–
(𝑥−ℎ)2
𝑏2
, where (h,k) is the center a is the units to move in the x-
direction from the center and b is the number of units to move in the y-direction from the center. The two
curves in the hyperbola open in the direction of the positive variable. The foci can be found using: a2 + b2 =
foci2 or denominator + denominator = foci2