11.1+Content+-+Conics+A

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.
 * Quick Instructions**


 * 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.

toc 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: > math {\left( {y - k} \right)^2} = 4p\left( {x - h} \right) math or: math y=\frac{1}{4p}a^{2}+bx+c math
 * It opens to the left or right
 * ‍‍‍‍The standard equation is:
 * 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: {\left( {x - h} \right)^2} = 4p\left( {y - k} \right) math Here is a sample parabola:
 * It opens up or down
 * The standard equation is:
 * math
 * 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

= = =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: math \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1 math , where math a^2>b^2 math math c=\sqrt{a^2-b^2} math from the center, so the foci are at (h-c,k) and (h+c,k)
 * The standard equation is
 * 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

If an ellipse has an y-orientation:

math \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 math where math a^2>b^2 math math c=\sqrt{a^2-b^2} math from the center, so the foci are at (h,k-c) and (h,k+c)
 * The standard equation is
 * 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

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: math \frac{(x-h)^2}{a^2}-\frac{(y-h)^2}{b^2}=1 math
 * 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

math c=\sqrt{a^2-b^2} math so the foci are (h-c,k) and (h+c,k) math y-k=\pm \frac{a}{b}(x-h) math To find the equations, you may also use the box method; however you don't have to draw these into the graph
 * The equations of the asymptotes are

If a hyperbola is a north/south hyperbola: math \frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1 math math y-k=\pm \frac{b}{a}(x-h) math
 * 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

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


 * Parabola:** eccentricity=1

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

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**= math \[\frac{(y-3)^2}{9}-\frac{(x-1)^2}{16}= 1\] math

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**= math \[(y+3)^2=-6(x-2)\] math

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**= math \[\frac{(x+3)^2}{25}+\frac{(y-8)^2}{100}= 1\] math

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.**

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