11.3+Content+-+Conics+B

Add Conics B 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**

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 * Conics (Unit 11B) Learning Targets **
 * To sketch the graph of semi-parabolas, semi-circles, semi-ellipses and semi-hyperbolas.
 * To write equations of conic sections or semi-sections of conics.
 * To solve systems of conic section equations.
 * To sketch the graph of conics in parametric form.

= = Equations are usually set up like this:

math \frac{b}{a} \sqrt{r^{2}-(x-Horizontal Shift)^{2}}+Vertical Shift math

-b/a represents the slope = = -the positive or negative sign under the radical is what determines if the graph will be a hyperbola or an ellipse -*if the slope is one, then the graph will be a cirlce You can either rearrange it to the standard conics form: math \frac{(x-horizontal shift)^2}{a^2}+\frac{(y-vertical shift)^2}{b^2}=1 math (Be aware that despite now being in the standard form, it still has the same limits as it did in the partial conics form.) Or graph from the above equation. You can tell which part to graph from whether it's positive or negative, and whether it's y= or x=.
 * || y= || x= ||
 * + || Top || Right ||
 * - || Bottom || Left ||

**Circles:**
The above graph is a circle, if math \frac{b}{a}= 1 math This means there's no stretching for the axis, and it can be graphed by finding the horizontal and vertical shift, and then using the radius to find the edges.

math \[y=\sqrt{25-(x+4)^{2}}-4\] math

= =

**Ellipses:**
When math \frac{b}{a}\neq 1 math , it's either a hyperbola or an ellipse. It's an ellipse if math \[\sqrt{r^{2}-x^{2}}\] math . The numerator is the **b** value, and the denominator is the **a** value, which is vertical and horizontal, respectively, if the equation is y=, or horizontal and vertical if the equation is x=. Find the center using the horizontal shift and vertical shift, and then use the a and b values to find the extent of the ellipse.

math \[y=\sqrt{25-y^{2}}+2\] math



**Hyperbola:**
Hyperbolas are also formed wh = = en math \frac{b}{a}\neq 1 math ‍Hyperbolas have a different base equation that circles or ellipses. Horizontal Hyperbolas (opening east and west) are formed from the equation ‍ math y=\frac{b}{a}\sqrt{\left ( x-horizontal shift \right )^{2}-r^{2}}+ vertical shift math Vertical hyperbolas (opening north and south) are formed from the equation math y=\frac{b}{a}\sqrt{r^{2}+\left ( x-horizontal shift \right )^{2}}+ vertical shift math These two equations are reversed for equations starting with x=. The section of the equation shown is the same as in the above chart. math \pm \frac{b}{a} math is the slope of the asymptotes of the hyperbola.
 * If the horizontal shift has a negative symbol in front of it, then the graph opens up in the opposite direction than expected. (i.e. If it's a y= hyperbola then it would be expected to open north south, but if there is a negative horizontal shift then the hyperbola actually opens east and west)

math y= \frac{2}{3}\sqrt{25+x^2} math = =

math y= \frac{2}{3}\sqrt{x^2-25} math



=__‍‍‍Quick Graphs for Conics: ‍‍‍__= = = This is the chart from our 11_06 class notes It shows various examples of quick graphs and information regarding the graphs! = =

**Let the graphs speak to you!!**
http://www.youtube.com/watch?v=U3n-NDpXCCc&list=FL9S-NQO0e3Y0q3oAVZc7LrA&index=1&feature=plpp_video (video made by Evan and Jennifer :) )
 * for example...**

**What does the equation mean?**

 * Well here is the given graph**

= =
 * And here is a lovely explanation of the equation!**

= = = = = = = = = = = = =__**‍Solving Systems of Conics: ‍**__=
 * Systems of conics can betricky, the best way to start a problem, such as the one below, is to start by looking for easy substitutions.
 * The best thing to do is simply to double check your algebra after every 3 steps or so.

__Steps for Tutorial Problem:__
> = = > will cancel.
 * 1) From the problem below, we can see that by solving for //y// on the bottom equation, we will end up with a square root of some sort. Luckily, the top equation has a //y^2,// so the radical
 * 1) Later, by substitution we end up with a quadratic, and after some algebra, we are able to solve for //x//.
 * 2) Using the //x// values we have just obtained, we can substitute those back into one of our original equations (preferably the bottom equation, it's much cleaner).
 * 3) Finally, after what seems like years of algebra, we are left with our solutions.

(1) math \[\left\{\begin{matrix}\frac{x ^{2}}{4}-\frac{y^{2}}{36}=1 \\ 2y^{2} -2=x \end{matrix}\right.\] math math \[2y^{2}=x+2\] math math \[y=\pm \sqrt{\frac{x+2}{2}}\] math (2) math \[\frac{x^{2}}{4}-\frac{x+2}{72}=1\] math math \[18x^{2}-x-2=72\] math math \[18x^{2}-x-74=0\] math math \[x=\frac{37}{18} \mathrm{ and } -2\] math (3) = = math \[2y^{2}-2=(\frac{37}{18}) \textup{ and } 2y^{2}-2=(-2)\] math math \[y=\pm \frac{\sqrt{73}}{6}\textup{ and }0\] math (4) math \[\textbf{Solutions:}(\frac{37}{18},\frac{\sqrt{73}}{6});(\frac{37}{18},-\frac{\sqrt{73}}{6});\textup{ and }(-2,0)\] math =__**‍‍‍Sketching the Graphs of Conics in Parametric Form:**__= Before we go on, it is important that you understand the basic rules and guidelines for basic parametric problems, as reviewed in 10.1. Click the link below if you need a refresher:

10.1 Content - Parametrics

Aside from reviewing unit 10, it may be a good idea to go back and look at unit 7, the topic being covered is highly associated with those Trig Identities that everyone loves. Below is the link to get to the list of Trig Identities: = = 7.1 Content - Verifying & Simplifying Using Trigonometric IDs

As you go about the next problem, have these identities in the back of you mind for reference.

__Steps for Tutorial Problem:__
1. Isolate for //t// in both equations We start this by simple algebra, but then we hit a wall. How are we supposed to use //t// if it's "trapped" inside a Trig function? Lucky for you, you know your Trig Identities extremely well and are able to remember that: math \[\csc ^{2}(t)-\cot ^{2}(t)=1\] math

2. Substitution -- And after some quick simplification, we have given birth to a beautiful hyperbola equation! (More info on graphing hyperbolas above) But we're not done just yet!

3. Now specify the motion of a particle that travels along the path of the graph pre-determined by our parametric equations (see steps for finding the motion of a particle without a t-restriction). To make this problem easier, there is no restriction. --However, there are problems in 11.4 that have t-restrictions thrown in.
 * This is where restrictions on //t// come into play.**

(1) = = math \[x=cot(t)\textup{ and }y=2csc(t)\] math math \[cot(t)=\frac{x}{3}\textup{ and }csc(t)=\frac{y}{2}\] math

(2) math \[(\frac{y}{2})^{2}-(\frac{x}{3})^{2}=1\] math math \[\frac{y^{2}}{4}-\frac{x^{2}}{9}=1\] math

(3) ==

__Steps for Finding the Motion of a Particle without a t-Restriction:__
1. Make an //x-t// and a //y-t// graph 2. Determine how the //x// and //y// variables reac = = t as //t// approaches ∞. 3. Using your //y-x// graph, follow the //x// and //y// values according to your guidelines found in step 2.

(1) math \[x=3cot(t)\] math

math \[y=2csc(t)\] math ‍‍‍ ‍‍‍ (2) math \[\textup{As }{t \to \infty}, x\textup{-values decrease from +}\infty \textup{ to 0, and then to -}\infty \] math math \textup{As }{t \to \infty}, y\textup{-values decrease from +}\infty \textup{ to 2, and then back to +}\infty \textup{. Then, }y\textup{-values decrease to -}\infty \textup{ at }t=\pi \textup{, increase to -2, and then go to -}\infty. math

(3) ==
 * The following steps are not applicable to this problem, they are merely hypothetical steps that you would take in the case of a restriction.**

__Steps for Finding the Motion of a Particle with a t-Restriction:__
1. Plug the lowest value for //t// into both of your original //x-t// and //y-t// equations 2. Follow the path of the particle as //t// increases (up to the highest value of the restriction).


 * These problems were found in your 11_08 notes, titled: "Systems & Parametrics".


 * Primary authors of this page (as of 06/02/12):**
 * Ari L.**
 * Chris S.**
 * Will L.**
 * Grace D.**
 * Evan A.**