I'm
going to make math as fun as a sunny day!!!HELP for anyone!!!
10.5
Conic Sections
Maria F
2nd period ICM
March 6th 2003
I'm
going to make math as fun as a sunny day!!!
What are conic sections? They are Circles, Points, lines, intersecting lines, hyperbolas, parabolas, and ellipses that occur when a plane and a cone intersect ......the general equation for this is:
Ax²+Bxy+Cy²+Dx+Ey+F=0 (A, B, C can not all be zero)
On this page I will help you figure out the conic section through the equation and I will help you find the eccentricity!!! The eccentricity is the ratio of the curve made by the conic section.
| Conic Section: | Standard Form of Equation: | Variation of General Form of Conic Equations: |
| circle | (x-h)²+(y-k)² = r² | A = C |
| parabola | (y-k)² = 4p (x-h) or
(x-h)² = 4p (y-k) |
Either A or C is zero. |
| ellipse |
(x-h)²/a² + (y-k)²/b² =1 or (y-k)²/a² + (x-h)²/b² =1 |
A and C have same sign, and A can't equal C. |
| hyperbola | (x-h)²/a² - (y-k)²/b² =1
(y-k)²/a² - (x-h)²/b² =1 or xy=k |
A and C have opposite signs. |
(Advanced Mathematical Concepts, Precalculus with applications, pg. 562)
This chart makes it a lot easier to handle different conic sections!
Problem :
The following points will make up a body of water in the west of Florida. A huge hole will be dug into the ground and they will fill it with water. Imagine that this body of water represents a conic section. Find out what conic section is represented and prove this with the table above. Then find the equation, and the eccentricity so that the builders can make the shape accurate!
a=(-3,3) b=(0,5) c=(0,3) d=(0,1) e=(3,3)
Answers:
STEP 1) Alright this problem looks harder then it really is!!! Lets graph the points...
STEP 2) Now we have to figure out the equation for this conic section...when we look at the graph the point make up an ellipse. The standard form of an Elipse is (x-h)²/a² + (y-k)²/b² =1 or (y-k)²/a² + (x-h)²/b² =1.
x and y represent the center. On our graph the x value is zero and the y value is three. There for the center is at (0,3).
The vertices are the two points in the direction were the elipse is pulled. Here points E and A are our vertices...V1= (-3,3) and V2= (3,3)
STEP 3) Since the elipse is pulled to the sides the x term has to come first...now we have all the information needed!
(x)²/9 + (y-3)²/4 = 1
-------> there we have our first answer! To check with the table of conics we have to have it in standard form. That is true and then we see that A and C are not equal but they have the same sign, there for we have an elipse!!!
STEP 4) Next step is to find out the eccentricity! The eccentricity for an elipse has to be between 0 or 1! The formula for eccentricity is the foci distance over the vertex distance.
STEP 5) We have the vertex distance which is six but then we need the foci! For this problem the foci is at
F1= (-¬5, 3)
F2= (¬5, 3)
-----> the foci distance is 4,5!
The eccentricity is 3/4 (0,75) this also proves that it is an elipse! Now the diggers of the hole know what to do and the body of water will have an elipse shape!!!
If you need any more help on conic sections and in particular the elipse then visit the following web site. It also includes various graphs and pictures that make conic sections more visual! http://www.math2.org/math/algebra/conics.htm
KINDNESS
No act of kindness, no matter how small, is ever wasted.
~Aesop~
This quote tells us that anything you do that helps others, builds there self esteem or makes them happy is worth it. It doesn't matter how small it is and even if the other person wouldn't notice, it might help them in another situation. You make yourself happy by helping others and even when you don't notice kindness is always a good thing!!!