Ellipses

*** ***** ******** ** ***********************

** **** * *** +36 - ** * 4y^2-24y * ** * ** * ** * ( ****** & *********** ** both after (9x^2 * **** *** 4y^2-24y *****

** * ** * * )^2 * ** * ( 2y - * *** * ** * ** * **

* (x + * )^2 + * ** * 3 *** = 36+ ** - 36 * *** So

(x * * *** * ( **** * + ** - * *** * (36 / * ) * * **** * ******** the

****** ** * ** * * ******* rotating *** ********* **** *** ******** *** *** coordinates with ******* to *** new axis by * & * ** **** Let * *

* * * *** * * * = Y ****** * Substituting these ****** ** eq * ** ***** * ** *

******* * Y^2 * (6/2)^2 = 1 * * ** * ***

* *** / *** * 1 *** **** ** the ********

***** represents ** ellipse **** is *** of *** ****

*** / a^2 + *** * *** * 1 ** the ***** ******** ** ******* **

**** * = * and b * 3 ******* * **** b so *** *****

***** ***** axis of *** ***** ******* *** along * *** * **** ************* ** ****** ** ***** **** = **

* * * * * 6 ****** ** minor **** is * * * * 2 x * * ** Vertices * The vertices ** *** ellipse

with ******* ** the new **** are * * * and * * b ** * ** * * ** *** * * 3 - 3 ** *** ******** **** ******* ** *** ***

**** ** ***** ** ( ***** * ) * = * - * * ** ** and * * * * * * * ie * ** * * *** * -2 ** ************ ** e ** ***** by √( 1

* ******* ) = *** * 1 * **** * ****** * 3 Coordinates ** **** *** ( be ) and

( -be * ** * * * **** * * * * * *** 5 ** coordinates of **** *** * **** * ***

( * √ 5 *

** *** at ******* * ******* ** **********

like ***** *** ******** *** ******* of completing squares ****

* *** * **** * *** + 36 * * (x^2

+ *** * 4 **** -6y) + ** * * *** *

**** - 4) * * *** * 3)^2 * 9) * ** = * ** * ****

* 36 * *** * 3)^2 - ** + ** = ****** *** isolated ********* **

******** 9 *** * 2)^2 *

* (y ***** * ** ******** **** ** *** * x

+ **** /4 *

(y * ****** * ** Thus ******* **** * *********** =

** The ****** ** *** ellipse **

***** and a * * * * * ***** **** *** ****** ** ****** of arc ************ is * and vertically **** ****** to ****** of *** ** 3 **** ******* ********* ********** **** ** 6 ** * **** ****** to ****** and from ** ** ****** to ****** along x-axis *** see ******** **** *** *******

Click here to download attached files: ellipse.docx