MTH 146 Conics Supplement

Transcription

1 105- Review of Conics MTH 146 Conics Supplement In this section we review conics If ou ne more detils thn re present in the notes, r through section 105 of the ook Definition: A prol is the set of points in plne tht re equidistnt from fix point F (the focus) nd fix line (the directrix) Eqution of Prol: An eqution of prol with focus 0, p nd directrix p is x 4p An eqution of prol with focus p,0 nd directrix x p is 4px Picture: A picture of ech kind of prol descri the ove equtions is elow: F F

2 Note: In ech cse the directrix is drwn thick nd the focus is lel The prol for the eqution with the x squr is opening up, nd the prol for the eqution with the squr is opening right Note: A derivtion of the formul for the eqution of prol is given on pge 671 of the ook Note: The vertex of the prol for oth of these eqution tpes is the origin Note: We llow p to e negtive Exmple: Find the vertex, focus nd directrix of the prol given x We cn rewrite this eqution s x Thus, we hve n eqution of the form 1 4px We note tht 4p p From this we conclude tht the 1 focus of this prol is,0 nd the directrix is 1 x Furthermore the vertex of the prol is 0,0 Definition: An ellipse is the set of points in plne the sum of whose distnces from two fix points (the foci) is constnt A picture of n ellipse is: (0,) (-,0) F 1 = (-c,0) center F 1 = (c,0) (,0) (0,-)

3 Terminolog: In the picture (this is tpicl ssumption for ellipses) We cll the points,0 nd,0 the vertices of the ellipse, nd we cll the line segment joining these two points the mjor xis Similrl the line segment joining the points 0, nd 0, is cll the minor xis The foci re lws on the mjor xis nd we lws use c to denote the distnce from the center to focus Similrl, is lws us to denote the distnce from the center to vertex Note: It is possile tht the mjor xis is verticl rther thn horizontl Eqution of n Ellipse: The ellipse given where c x where 0,0 1 hs foci c,0 nd vertices Its center is 0,0 The ellipse given where c x where 0 0, 1 hs foci 0, c nd vertices Its center is Exmple: Find n eqution of the ellipse with foci 0, nd vertices 0, 3 0,0 Bs upon the loction of the vertices, we know tht the center of the ellipse is 0,0, nd we know tht the mjor xis is verticl We lso know tht the distnce from the center to vertex is 3 So, 3 Also the distnce from the center to focus is So, c B wht is given ove, this mens 3 5 Thus, the eqution of the ellipse is x x Grd Exmple: () Find the vertices nd foci of the ellipse given Solution: We see tht 36 nd 8 Since the ppers under the x in the eqution we know tht the mjor xis of the ellipse is horizontl Furthermore, the ove equtions we know tht the vertices re 6,0 To find the foci, we must find c We know tht c This mens tht 7 c nd the foci re 7,0 () Grph the ellipse

4 Definition: A hperol is the set of ll points in plne the difference of whose distnces from two fix points (the foci) is constnt A picture of hperol is: Terminolog: In the picture c,0 nd,0 the vertices of the hperol, nd we cll the line segment joining these two points the trnsverse xis The foci re c,0 nd c,0, nd the foci re lws on the trnsverse xis nd we lws use c to denote the distnce The lines drwn in green re the slnt smptotes of the hperol The point t which the slnt smptotes intersect is the center of the hperol (this lws holds for hperol) We cll the points Note: It is possile tht the trnsverse xis is verticl rther thn horizontl

5 Eqution of Hperol: The hperol given c x, vertices,0 Its center is 0,0 hs foci c,0 1, nd slnt smptotes where x The hperol given c x, vertices 0, center is 0,0 hs foci 0, c 1, nd slnt smptotes where x Its Grd Exmple: Find the eqution nd slnt smptotes of the hperol with vertices 0, 0, 5 nd foci Solution: Since the vertices re 0, we know tht the trnsverse xis is verticl nd the center of the hperol is the origin Moreover, we know tht Similrl, since the foci re 0, 5, we know tht c 5 B the ove equtions we know tht c Now, the eqution of the hperol is the equtions of the slnt smptotes re Note: In hperol we don t ne x 1, ut in n ellipse we ne x 1 nd 4 1 Fct: The eqution representing conic tht hs een shift h units horizontll nd k units verticll cn e form replcing nd x s in the originl eqution with x h nd replcing n s in the originl eqution with k Exmple: Find n eqution for the prol with vertex which contins the point 1,5,3, xis of smmetr x, nd Bs upon the given we know tht this prol opens upwrd From lst clss we know tht if its vertex ws t the origin, its eqution would e of the form x 4p Since its vertex is t,3 we know tht the desir eqution is of the form x 4p 3 We lso know tht 1,5 must stisf this eqution Thus,

6 we hve 1 1 4p p p Thus, the desir eqution is 8 x 3 1 Note: The ook will usull refer to the xis of smmetr s eing verticl or horizontl nd simpl stte verticl/horizontl xis So, the ook would write the ove prolem s: Find n eqution for the prol with vertex,3, verticl xis, nd which contins the point 1,5 Exmple: Find the vertex/vertices of the conic determin the eqution 9x 4 7x The ke in question like this is to complete the squre in order to get the eqution in form tht we cn recognize Specificll, we notice: 9x 4 7x x 7x x x x x x 1 x From this we recognize tht the eqution determines hperol with verticl trnsverse xis nd center 4,1 Since 3, we conclude tht the vertices re 4,4 nd 4, Grd Exmple: Find n eqution of the ellipse with foci,, 4, 1,, 5, nd vertices Solution: We know tht the center of the ellipse hs to e t the midpoint of the mjor xis Thus, the center is 3, Also we see tht nd c 1 Since c, we know tht 3 Thus, the desir eqution is x

7 106 Conic Sections in Polr Coordintes Theorem: Let F e fix point (the focus) nd l e fix line (the directrix) in plne Let e e fix positive numer (the eccentricit) The set of ll points P in the plne such tht PF Pl e (where Pl denotes the distnce from the point P to the line l ) is conic section The conic is n ellipse if e 1, prol if e 1, nd hperol if e 1 Proof: B the definition of prol from section 105, we know tht if e 1, the set of ll PF points P in the plne such tht 1, is prol Pl Now, suppose e 1 Since we could plce n x nd xis on the plne fter we know where F nd l re loct, it is ok to view F s eing t the origin nd l s eing prllel to the -xis Suppose P is n ritrr point which stisfies PF Pl e Let r, e the polr coordintes of the point P, nd let x d e the eqution which determines the line l (for now we will ssume tht d 0 ) A picture of the sitution is elow: P r x = d F We note tht PF r nd Pl d r cos Since PF e PF e Pl, we know tht Pl r e d r cos r e d r cos

8 Now, if we convert to rectngulr coordintes nd mnipulte the eqution we otin: x e d x x e d dx x x e d de x e x de 1 e x de x e d 1 e x x e d 1 e de d e d e 1 e x x e d 1e 1 e 1e 4 4 de e d de 1 e 1 e 1 e 1 e 1 e 1 e x x de x 1 e 1 1 e 1 e e d We note tht if e 1, then this eqution determines hperol, nd if e 1, then this eqution determines n ellipse A similr rgument shows the desir when d 0 nd when d 0 Note: If the eccentricit is not 1, then we hve n ellipse or hperol nd the eqution holds c e Question: Wht hppens to the conic s the eccentricit pproches 0? (Answer: The conic strts to look more nd more like circle) Note: From the prt of the proof where e does not equl one we see tht polr eqution for the conic is r 1 ecos This ls us to the next theorem Theorem: A polr eqution of the form r 1 ecos or determines conic section with eccentricit e r e 1 sin Proof: The proof cn e done emploing the ide us in the ove proof

9 10 Grd Exmple: Consider the polr eqution: r 3 cos () Find the eccentricit of the conic determin this eqution Solution: We notice tht know tht e r 3 cos 1 cos 3 Then, from the ove theorem, we () Wht tpe of conic does this eqution determine? Solution: Since the eccentricit is less thn 1 the eqution determines n ellipse (c) Use our clcultor to sketch picture of the conic determin this eqution Solution: Fct: The following tle will help one identif the directrix of conic which is given polr eqution Eqution Form Directrix Eqution r x d 1 ecos r x d 1 ecos r d e 1 sin r d 1 esin

10 Grd Exmple: 10 () Find the eqution of the directrix of the ellipse determin r 3 cos Solution: We lr know tht r This mens 3 cos 1 cos d d 5 So, the ove tle the directrix is x () Suppose the ellipse from prt () is rott through n ngle of 60 out the origin, find polr eqution of the resulting ellipse Solution: To find such n eqution we ne onl replce with 10 r 3cos 3 Recommend Homework: 105: 1; 3; 11; 13; 19; 1; 5-47 odd; 61; : 1-19 odd So, n eqution is 3