10-3 Ellipses
Danielle D.
ICM Honors
John is an astronomer who likes to look at many galaxies at night. One night, he found this strange galaxy. The next day, he asked his friend for the exact location of the galaxy. He said he didn't know, but he did have the equation for the new galaxy. Therefore, John's friend gave him the equation: 2x2+y2-4x+6y+9=0. In order for John to find the exact equation of the galaxy he needs to find the standard form of the equation, the center, foci, and vertices.
2x2+y2-4x+6y+9=0 <---the equation John was given
2x2-4x+y2+6y=-9 <---put your X's and Y's together (X's first!)...the squared variables first, and the left over number on the right side of the equation.
2(x2-2x+1)+(y2+6y+9) = -9+2(1)+9 <---then complete the square! to do this, for example, take the 2 out from all the X's...so now you will have 2(x2 -2x....then half the two (which equals one) and square it! Now your complete equation is 2(x2-2x+1)...do the same for the Y's!! Then add the terms you took out....the two, times the one (the halved then squared term) and the 9.
2(x-1)2
+ (y+3)2 = 2 <---now it should equal 2! Divide
all the terms by 2!
2
2 2
(x-1)2+ (y+3)2= 1 <---the standard form
of the equation!
1 2
Therefore, the center is at (1,-3) (which are the numbers after your X and Y...with changed signs)
a=1 (the denominator in the first fraction), b=2 (the denominator in the second fraction) and it is a fact that c2 = a2 - b2!
Plug in the
"a" (which equals one), and the "b" which equals
2...therefore
c= the square root of 3!
Since the center is at (1, -3), you need to translate each point one unit to the right, and three units down.
With this information, you now know the foci is (1, the square root of 3 subtracted by 3)
and (1, the negative square root of 3 subtracted by 3).
The vertices are now (2,-3), (0,-3), (1,-1), and (1,-5).
With the information John now has (the standard form, center, foci, and vertices) he knows where his galaxy is located when he looks up at the stars at night!
| Henry David Thoreau | If a
man does not keep pace with his companions, perhaps it is because he hears
a different drummer. Let him step to the music which he hears, however
measured or far away.
~RESPECT~This quote means that one needs to respect another's beliefs, thoughts, and ideas, no matter how different they may be. |
Go to this fun site! ---> http://www.math.hmc.edu/funfacts/
Quote is from: http://quotes.telemanage.ca/quotes.nsf/QuotesByCat!ReadForm&Start=1&Count=2000&Expand=89
Pictures are from:
http://www.bellsnwhistles.com/asta/ast013.gif
