Ellipses The seond type of oni is lled n ellipse. Definition of Ellipse An ellipse is the set of ll points (, y) in plne, the sum of whose distnes from two distint fied points (foi) is onstnt. (, y) d d Fous Fous d + d is onstnt. There re numer of prts of n ellipse tht should e noted:
overte verte mjor is Fous enter Fous verte minor is overte The midpoint etween the foi is the enter. The line segment through the foi, with endpoints on the ellipses, is the mjor is. The endpoints of the mjor is re the verties of the ellipse. The line segment through the enter nd perpendiulr to the mjor is, with endpoints on the ellipse, is the minor is. The endpoints of the minor is re the overties of the ellipse.
There re 3 distnes tht re importnt when studying n ellipse. Fous Fous Fous Fous is lwys the longest length nd is lwys the fol length 3
Stndrd Eqution of n Ellipse The stndrd form of the eqution of n ellipse entered t (h, k), with mjor is of length, minor is of length, where 0 < <, is ) ( ) ( k y h Mjor is is horizontl ) ( ) ( k y h Mjor is is vertil The foi lie on the mjor is, units from the enter, with. If the enter is t the origin (0, 0), the eqution tkes one of the following forms. y Mjor is is horizontl y Mjor is is vertil
Emple: Find the enter, verties, nd foi of the ellipse given y 9 + y = 36. Solution: First divide through y 36 to get the orret form. y 9 Rememering tht the lrger numer on the ottom orresponds to, we n see tht: so 9 so 3 Using, we n find. 3 The enter of the ellipse is (0, 0). From tht point: We go 3 units up nd down for the verties: (0,3), (0,-3) We go units right nd left for overties: (, 0), (-, 0) We go units up nd down for the foi: (0, ), (0,- )
- - Emple: Find the stndrd form of the eqution of the ellipse entered t the origin with mjor is of length 0 nd foi t (±3, 0). Solution: If the mjor is is 0, we know tht =. If the foi re t (±3, 0), we know tht = 3. Solve for. 3 6 6
Sine the foi lwys lie on the mjor is, we know tht the mjor is is horizontl. Tht tells us tht goes under the term. Sine it is entered t the origin, we hve: y 6 y - - 7
Emple: Find the stndrd form of the eqution of the ellipse with foi (0, 0) nd (0, ) with mjor is of length of 8. Solution: If the mjor is is 8, we know tht =. If the foi re t (0, 0) nd (0, ), we know tht is hlf the distne etween them so =. Solve for. 3 Sine the foi lwys lie on the mjor is, we know tht the mjor is is vertil. Tht tells us tht goes under the y term. The enter is hlf wy etween the foi, so the enter must e (0, ). Thus we hve ( 0) ( y ) 6 8
y - - Emple: Sketh the grph of y 6 8y 9 0. Solution: You need to omplete the squre with the -terms nd the y-terms. y 6 8y 9 0 *Get the onstnt on the other side nd group the -terms together nd the y-terms terms, putting in lnks on oth sides of the eqution. 9
( 6 ) (y 8y ) 9 *Before ompleting the squre, pull the out of the y group. ( 6 ) ( y y ) 9 *Complete the squres. ( 6 9) ( y y ) 9 9 *Rememer to dd () on the right for the y group. ( 3) ( y ) *Beuse we need on the right, divide through y. ( 3) ( y ) ( 3) ( y enter: (-3, ) mjor is: horizontl nd = minor is: vertil nd = ) 0
y - - Emple: Find the enter, verties, nd foi of the ellipse 6 9y 96 36y 36 0 Solution: Put the eqution in stndrd form y ompleting the squres.
6 (6 6( 6( 6( 3) 6( 3) ( 3) 9 9y 96 6 96 36y ) ) 9( y 6 9) 9( y 9( y ( y ) 6 ) (9y 9( y ) y 36 36y y enter: (3, -) mjor is: vertil nd = minor is: horizontl nd = 3 0 ) ) ) 36 36 36 36 Find the foi y first finding. 7 7 3 The foi re 7 units ove nd elow the enter, so the oordintes would e (3, -+ 7 ) nd (3, -- 7 ).
y - - Eentriity To mesure the ovlness of n ellipse, we use the onept of eentriity. Definition of Eentriity The eentriity e of n ellipse is given y the rtio e 3
*Note: Beuse is lwys greter thn, the frtion will lwys e etween 0 nd. When is lose to, tht mens the foi re lose the the verties nd the frtion is lose to. When is muh smller thn, tht mens the foi re lose to the enter nd the frtion is smll.
Emple: Find the eentriity of the ellipse ) ( 9 ) ( y Solution: Find. 6 3 The eentriity is e Emple: Find the eentriity of the ellipse 7) ( ) ( y so e e e
Applition Emple: A pssgewy in house is to hve stright sides nd semielliptilly-rhed top. The stright sides re feet tll nd the pssgewy is 7 feet tll t its enter nd 6 feet wide. Where should the foi e loted to mke the templte for the rh? 6 = 3 nd =. Find. 3.36 The foi should e pled.36 feet to the right nd left of the enter of the semiellipse. 6