Name ________________________________
Introduction to Conic Sections
This introduction to conic sections is going to focus on what they
some of the skills needed to work with their equations and graphs.
year, we will only work with circles and parabolas.
Center
axis
are and
This
Physical Definitions
A double cone is formed when a line is revolved around an axis. All
conic sections come from the shape exposed when you cut off a
section of a cone. How you cut the cone determines the shape.
revolving line
Parabola: The slice is made by cutting parallel to the revolving line.
Circle:
The slice is made by cutting perpendicular to the center axis.
Ellipse:
The slice is made by cutting more shallowly than the revolving line (won’t go through the
bottom).
Hyperbola: The slice is made by cutting more steeply than the revolving line (will go through the
bottom).
Conic
sections
also
called
are
quadratic relations. The standard form of a quadratic relation is
Ax 2 + Bxy + Cy 2 + Dx + Ey = F
For most of our work, coefficient B will be zero. This is, in general, the mechanism that rotates the
conic sections and will be studied in more depth in a later course.
It should be noted that most conic sections are not functions, only relations.
Name_____________________________________
Honors Math 2
Geometric (Locus) Definition of a Parabola
A parabola is a locus defined in terms of a fixed point, called the focus, and a fixed line, called the
directrix. A parabola is the set of all points P(x, y) whose distance from the focus (F) equals its
distance from the directrix. In other words, PF = PD (where D is the point on the directrix closest to
P – so PD is perpendicular to the directrix).
The figure shows a parabola with its focus at
(0, 1) and a directrix of y = −1 . Equally
spaced concentric circles with their center at
the parabola’s focus enable you to measure
distances from the focus. Equally spaced
€ parallel to the parabola’s
horizontal lines
directrix enable you to measure vertical
distances from the directrix. This type of
graph paper is called focus-directrix graph
paper.
6
4
P
6
6
2
F
–5
Notice that P is the point of intersection of
the circle centered at (0, 1) with a radius of 6
and the horizontal line 6 units above the
directrix. Thus, P is equidistant from the
focus and directrix. Examine the figure and
note that all points on the parabola are
equidistant from the focus and the directrix.
5
D
–2
–4
–6
10
Example:
In the diagram, a focus has been
drawn at the point (0, 3) and a
directrix has been drawn, y = –3.
8
6
a. Plot the points that are
equidistant from the focus and
directrix using the concentric circles
and the grid.
4
F
2
b. Identify the vertex of the
resulting parabola.
– 10
–5
5
–2
c. Identify p, the distance from the
focus to the vertex (and the distance
from the directrix to the vertex).
–4
–6
–8
– 10
10
Deriving the Algebraic Definition of a Parabola
For now, let’s look at parabolas whose focus is the point (0, p) and whose directrix is y = – p. This will mean
the vertex of the parabola is at the origin, as shown in the sketch below.
P (x, y)
If PF = PD, derive the standard equation for any point P on this
parabola using the distance formula.
F (0, p)
y = -p
D (x, -p)
What if the directrix is above the focus? Draw a picture similar to the one above. Derive the standard
equation for this case.
What if the directrix is a vertical line rather than a horizontal one? Consider both cases.
1
Example: Graph x = − y 2 and identify the vertex, focus and directrix.
8
€
Example: Write the standard equation of a parabola with its vertex at the origin and with the directrix y = 4.
If you need to graph a horizontal parabola on your calculator, solve for y and entered both equations into your
calculator.
Example: Solve for y, put the following relation in
your calculator, and sketch on the grid.
y 2 + 4x = 0
HOMEWORK
For the following three problems:
a. Plot the points that are equidistant from the focus and directrix using the concentric circles and the grid.
b. Identify the vertex of the resulting parabola.
c. Identify p, the distance from the focus to the vertex (and the distance from the directrix to the vertex).
d. Write the equation of the parabola.
10
1. Focus at (0, –3)
Directrix of y = 3
8
6
4
2
– 10
–5
5
10
5
10
–2
F
–4
–6
–8
– 10
10
2. Focus at (3, 0)
Directrix of x = –3
8
6
4
2
– 10
F
–5
–2
–4
–6
–8
– 10
3. Focus at (–3, 0)
Directrix of x = 3
10
8
6
4
2
– 10
–5
F
5
10
–2
–4
–6
–8
– 10
For the next three problems: Write the standard equation for the parabola with the given characteristics.
4. Given its focus at (1,0) and
5. Given its vertex at (0, 0)
6. Given its vertex at (0, 0)
its directrix at 𝑥 = −1
and its focus at (0, -5)
and its directrix x = 2
7. Graph x =
€
1 2
y and identify the vertex, focus and directrix.
20