Quick Instructions
Add Conics A content to this page. See specific details on the home page. This unit has been split into parts A & B. Content between the two must be cohesive as it is one continuous, connected material. They are split simply to better organize the page.


Conics (Unit 11A) Learning Targets
  • To sketch the partial graph of parabolas, circles, ellipses and hyperbolas.
  • To locate foci, directrix, and asymptotes and find the eccentricity of conics.
  • To write equations of conic sections or semi-sections of conics.


Conic section

‍1.Parabolas

A parabola is a set of points that are equidistant from a given point (focus) and a given line (directrix).

If a parabola has an x-orientation:
  • It opens to the left or right
  • ‍‍‍‍The standard equation is:
or:

  • The focus is (h+p,k) and the directrix is x = h-p
  • The vertex is (h,k)
  • ‍The common distance from the parabola to the focus and the directrix is p * The focal diameter, which is the distance between the two points on the parabola that form a line perpendicular to the axis of symmetry through the focus, is 4p.
  • If p>0, the parabola opens to the right; if p<0, it opens to the left.

If a parabola has a y-orientation:
  • It opens up or down
  • The standard equation is:
  • The focus is (h,k+p) and the directrix is y=k+p
  • ‍‍‍‍The vertex is (h,k)
  • The common distance from the parabola and the focus and the directrix is p.‍‍‍‍
  • If p>0 then the parabola opens up; if p<0 then the parabola opens down
Here is a sample parabola:
external image ParabolaFig9.gif


2‍. Ellipses

An ellipse is the set of points whose distances from two given points (foci) sum to a constant.
If the larger denominator in an equation of an ellipse is under the x-term, the ellipse has an x-orientation. If the larger denominator is under the y-term, the ellipse has a y-orientation.
If an ellipse has an x-orientation:
  • The standard equation is

, where

  • The major axis is parallel to the x-axis
  • The center is (h,k)
  • The vertices are (h-a,k) and (h+a,k), the endpoints of the major axis. So the minor axis is 2a units long.
  • The endpoints of the minor axis are (h,k-b) and (h,k+b)
  • The foci are always on the major axis. Each focus is at the distance of

from the center, so the foci are at (h-c,k) and (h+c,k)

If an ellipse has an y-orientation:

  • The standard equation is

where

  • The major axis is parallel to the y-axis
  • The center is (h,k)
  • The vertices are (h,k-a) and (h,k+a), the endpoints of the major axis.
  • The endpoints of the minor axis are (h-b,k) and (h+b,k)
  • The foci are on the major axis. Each focus is at a distance of

from the center, so the foci are at (h,k-c) and (h,k+c)

Sample graph:
external image ellipse-image013.gif


3. ‍Hyperbolas

Remember to talk to the hyperbolas!

A hyperbola is the set of pints whose distances from two fixed points (foci) differ by a constant. A hyperbola has two halves with asymptotes.
If the x-term is positive, the the hyperbola is a east west hyperbola. If the y-term is positive, then the hyperbola has a north/south orientation.

If a hyperbola is a east/west hyperbola:
  • The standard equation is

  • The center is (h,k)
  • The vertices are (h-a,k) and (h+a,k). The axis is horizontal and has a length of 2a.
  • The foci are always on the major axis of the hyperbola's equivalent ellipse. The distance between the center and each focus is


so the foci are (h-c,k) and (h+c,k)
  • The equations of the asymptotes are

To find the equations, you may also use the box method; however you don't have to draw these into the graph

If a hyperbola is a north/south hyperbola:
  • The standard equation is

  • The center is (h,k)
  • The vertices are (h,k-a) and (h,k+a)
  • The foci are (h,k-c) and (h,k+c)
  • The equations of the asymptotes are




Sample hyperbola graph (east/ west):
external image Hyperbola7.jpg

‍4. Eccentricity


The eccentricity of an ellipse or hyperbola is a measure of it's elongation. ‍‍‍‍Based off of eccentricity, one will know whether the shape will be a circle, ellipse, parabola, or hyperbola.‍‍‍‍



Eccentricity is calculated by c/a.

Circle: eccentricity=0

Ellipse: eccentricity of less than 1 since c < a.
The closer c is to a the more circular the ellipse.

Parabola: eccentricity=1

Hyperbola: eccentricity is greater than 1 since c > a
The farther c is from a, the more elongated the hyperbola.

For a cool interpretation of what eccentricity is in its application, click here!
This picture should show how the conics are formed from a 3D cone.
Now, you should notice that a plane cuts the cone a certain way. The interesting thing is that it seems like the eccentricity is the slope of the plane.
For example, if the eccentricity is 0, the conic formed is a circle. If you look at the image, you will see the plane is horizontal when a circle is formed, which means the slope of the plane is 0, just like the eccentricity. This is the same for every other conic. If a parabola is formed, it seems like the slope of the plane is 1, just like the eccentricity for a parabola. If a ellipse is formed, the slope of the plane is less than the slope of the plane when a parabola is formed, which means the slope must be less than 1, just like the eccentricity for a parabola. If a hyperbola is formed, the plane has a slope greater than that of the parabola, which means the slope is greater than one. That is the same for it's eccentricity.

‍‍‍‍Sample Problems‍‍‍‍ (snswers are further down)


Example 1



Which conic section does this equation define? Also find, if they exist,

(a) the center
(b) the vertex/vertices
(c) the focus and foci
(d) the directrix
(e) the asymptotes
(f) the eccentricity

Example 2




Which conic section does this equation define? Also find, if they exist,

(a) the center
(b) the vertex/vertices
(c) the focus and foci
(d) the directrix
(e) the asymptotes
(f) the eccentricity

Example 3




Which conic section does this equation define? Also find, if they exist,

(a) the center
(b) the vertex/vertices
(c) the focus and foci
(d) the directrix
(e) the asymptotes
(f) the eccentricity

Answer to Example 1

This is a hyperbola opening north/south.
a. The center is (1,3)
b. The vertices are (1,0) and (1,6)
c. The foci are (1,-2) and (1,8)
c=sqrt(3^2 +4^2)= 5
d. Does not exist
e. The asymptotes are y-3 =(+/-)3/4(x-1)
(Use the box method!)
f. The eccentricity is 5/3

Answer to Example 2

This is a parabola opening left.
a. Does not exist
b. ‍‍‍‍The vertex is (2,-4)‍‍‍‍
c. The focus is (1/2, -4)
-6=4p
p=-3/2
d. The directrix is x=7/2
e. Does not exist
f. Does not exist

Answer to Example 3

This is an ellipse with a y-orientation
a. The center is (-3,8)
b. The vertices are (-3,-2) and (-3,18)
c. The foci are at (-3,8- sqrt75) and (-3, 8+sqrt75)
c=sqrt(10^2-5^2) = sqrt75 = 5 sqrt 3
d. Does not exist
e. Does not exist
f. The eccentricity is (sqrt3)/2


Primary authors of this page (as of 06/02/12):
Grace D., Will H., Ari L.