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.

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:

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:

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):

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.

Quick InstructionsAdd 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## Table of Contents

A parabola is a set of points that are equidistant from a given point (1.Parabolasfocus)and a given line(directrix).If a parabola has an x-orientation:

- It opens to the left or right
- The standard equation is:

or: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:## 2.

An ellipse is the set of points whose distances from two given points (Ellipsesfoci) 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:

, where

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

If an ellipse has an y-orientation:

where

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

Sample graph:

## 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:

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

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:

Sample hyperbola graph (east/ west):

## 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=0Ellipse:eccentricity of less than 1 since c < a.The closer c is to a the more circular the ellipse.

Parabola:eccentricity=1Hyperbola:eccentricity is greater than 1 since c > aThe 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 1Which 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 2Which 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 3Which 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

This is a hyperbola opening north/south.Answer to Example 1a. 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

This is a parabola opening left.Answer to Example 2a. 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

This is an ellipse with a y-orientationAnswer to Example 3a. 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.##