Conic Sections
Conic sections are shapes that can be formed by the intersection of a plane and a cone. You can find a handy diagram of them
in the 3D Geometry Chapter 2 Section 6 of this book. There are four types of conic sections—circles, parabolas, ellipses, and
hyperbolas—and each can be represented in the Cartesian plane as a graph with a corresponding equation. By looking at the
equation, you can determine many things about the graph.
Equation
Parabola
y = a (x – h) + k
2
Interpretation
Vertex: (h, k)
Stretch factor: a
Axis of symmetry: x = h
Example
axis of summetry
Conic Section
0
vertex
y = (x – 2) 2 – 3
y
Circle
(x – h) 2 + (y – k) 2 = r 2
Radius: r
Center: (h, k)
(1, 0)
(x – 1) 2 + y 2 = 9
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x
y
Ellipse
^ x – h h2
a
2
+
(y – k) 2
=1
b2
Minor Axis
Horizontal axis: 2b
Vertical axis: 2a
Center: (h, k)
The shorter of the two
axes is called the minor
axis. The longer of the two
axes is called the major
axis. The semiminor axis
is half of the minor axis,
and the semimajor axis is
half of the major axis.
Major Axis
x
2
x 2 ( y – 1)
+
=1
9
4
(0, 1)
Center
0
tes
(y – k) 2
=1
b2
pto
ym
pto
–
tes
As
a
2
ym
^ x – h h2
Transverse
Axis
As
Hyperbola
(horizontal
transverse axis)
Distance between
vertices: 2a
Center: (h, k)
Asymptotes:
b
y = ! a ( x – h) + k
Transverse axis: y = k
2
x 2 (y – 1)
–
=1
4
9
(0, 1)
Center
0
tes
(x – h) 2
=1
a2
mp
pto
–
tot
ym
b
2
es
As
^ y – k h2
Transverse
Axis
y
As
Hyperbola
(vertical
transverse axis)
Distance between
vertices: 2b
Center: (h, k)
Asymptotes:
a
y = ! b ( x – h) + k
Transverse axis: x = h
(y – 1) 2
– x2 = 1
4
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Term
Focus
Definition
Interpretation
A point of reference used
to define conic sections,
particularly ellipses and
hyperbolas.
An ellipse is the set of
points for which the sum
of the distance from each
point to both foci is a
constant, and a hyperbola
is the set of points for
which the difference of
the distance from each
point to both foci is a
constant.
Example
P
F1
Foci
F2
Foci
Essential Technique: Completing the Square
You’ll often need to use a process called completing the square to simplify equations of conic sections, to eliminate linear
terms (terms that aren’t raised to any exponent), or to make your equation look like one of the standard forms of a conic
section shown in the chart above.
For example, you are given the equation of a circle, x2 + 6x + y2 – 4y = 3 and asked to find it's radius. At first glance, this
equation may not even look like a circle — but you can convert it to that form by completing the square.
1: Group the x and the y terms together.
(x2 + 6x) + (y2 – 4y) = 3
2: For each bracket, take the coefficient of the linear term,
divide it by two, square the result, and then add and subtract
this number in each bracket.
(x2 + 6x + 9 – 9) + (y2 – 4y + 4 – 4) = 3
3: Move the subtracted numbers to the other side of the
equation.
(x2 + 6x + 9) + (y2 – 4y + 4) = 16
4: Factor each bracket as a perfect square trinomial.
(x + 3)2 + (y – 2)2 = 16
You’re done! This equation is now in the standard form for the equation of a circle. You can see that the center of the circle
is (–3, 2) and the radius is 16 = 4.
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