H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.


 Reginald O’Connor’
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1 Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining qudrti eqution in. Eplin wh. (D) Show tht. u v (u ) (E) Show tht u v 4. Eplin wh this implies tht u v 4, with equlit if nd onl u if u v. () Whih fol hord is the shortest? Is there longest fol hord? (G) Is onstnt for fol hords of the ellipse? or fol hords of the hperol? Otin evidene u v for our nswers onsidering speifi emples. (H) The oni setion with fous t the origin, diretri the line D, nd eentriit E hs the DE polr eqution r. Eplin how this polr eqution mkes it es to show tht u v E os for prol. Use the polr eqution to determine the sum for fol hord of n ellipse or u v hperol. Chpter Review  CONIC SECTIONS; ARABOLA The plne urves otined interseting right irulr one with plne re lled oni setions. If the plne uts ler through one nppe, then the intersetion urve is lled irle if the plne is perpendiulr to the is nd n ellipse if the plne is not perpendiulr to the is. If plne uts onl one nppe, ut does not ut ler through, then the intersetion urve is lled prol. If plne uts through oth nppes, ut not through the verte, the resulting intersetion urve is lled hperol. A plne pssing through the verte of the one produes degenerte oni point, line, or pir of lines. The figure illustrtes the four nondegenerte onis. Cirle Ellipse rol Hperol
2 83 Additionl Topis in Anlti Geometr The grph of A B C D E where A, B, nd C re not ll, is oni. The following is oordintefree definition of prol: rol. 4 Verte: (, ) ous: (, ) Diretri: Smmetri with respet to the is Ais the is A prol is the set of ll points in plne equidistnt from fied point nd fied line L in the plne. The fied point is lled the fous, nd the fied line L is lled the diretri. A line through the fous perpendiulr to the diretri is lled the is, nd the point on the is hlfw etween the diretri nd fous is lled the verte. (opens down) (opens up) L V(Verte) d d d d Ais (ous) rol  ELLISE The following is oordintefree definition of n ellipse: Ellipse rom the definition of prol, we n otin the following stndrd equtions: Stndrd Equtions of rol with Verte t (, ). 4 Verte: (, ) ous: (, ) Diretri: Smmetri with respet to the is Ais the is Diretri An ellipse is the set of ll points in plne suh tht the sum of the distnes of from two fied points in the plne is onstnt. Eh of the fied points, nd, is lled fous, nd together the re lled foi. Referring to the figure, the line segment VV through the foi is the mjor is. The perpendiulr isetor BB of the mjor is is the minor is. Eh end of the mjor is, V nd V, is lled verte. The midpoint of the line segment is lled the enter of the ellipse. V d d Constnt B d B d V rom the definition of n ellipse, we n otin the following stndrd equtions: (opens left) (opens right)
3 Chpter Review 83 Stndrd Equtions of n Ellipse with Center t (, ). interepts: (verties) interepts: oi: (, ), (, ) Mjor is length Minor is length 3 HYERBOLA The following is oordintefree definition of hperol: Hperol A hperol is the set of ll points in plne suh tht the solute vlue of the differene of the distnes of to two fied points in the plne is positive onstnt. Eh of the fied points, nd, is lled fous. The intersetion points V nd V of the line through the foi nd the two rnhes of the hperol re lled verties, nd eh is lled verte. The line segment VV is lled the trnsverse is. The midpoint of the trnsverse is is the enter of the hperol. d d Constnt d d V V. interepts: interepts: (verties) oi: (, ), (, ) Mjor is length Minor is length rom the definition of hperol, we n otin the following stndrd equtions: Stndrd Equtions of Hperol with Center t (, ). interepts: (verties) interepts: none oi: (, ), (, ) Trnsverse is length Conjugte is length [Note: Both grphs re smmetri with respet to the is, is, nd origin. Also, the mjor is is lws longer thn the minor is.]
4 83 Additionl Topis in Anlti Geometr. h. h k k where re the oordintes of the origin reltive to the originl sstem. (, ) (, ). interepts: none interepts: (verties) oi: (, ), (, ) Trnsverse is length Conjugte is length (, ) (, ) Tle lists the stndrd equtions for trnslted onis. 5 ARAMETRIC EQUATIONS A plne urve is the set of points (, ) given the prmetri equtions f(t) nd g(t) where the prmeter t vries over n intervl I. The pth of projetile with n initil speed v t n ngle with the horizontl is given [Note: Both grphs re smmetri with respet to the is, is, nd origin.] 4 TRANSLATION O AXES In the lst three setions we found stndrd equtions for prols, ellipses, nd hperols loted with their es on the oordinte es nd entered reltive to the origin. We now move the onis w from the origin while keeping their es prllel to the oordinte es. In this proess we otin new stndrd equtions tht re speil ses of the eqution A C D E, where A nd C re not oth zero. The si mthemtil tool used is trnsltion of es. A trnsltion of oordinte es ours when the new oordinte es hve the sme diretion s nd re prllel to the originl oordinte es. Trnsltion formuls re s follows: (v os )t nd (v sin )t 4.9t, t or, fter eliminting the prmeter t, 4.9 (tn ) v os where t is time in seonds nd nd re distnes in meters. The rnge of projetile is the distne from the point of firing to the point of impt. If the initil speed v is held onstnt nd the ngle is vried, then the rehle region of the projetile is seprted from the nonrehle region prol lled n envelope of the possile proli pths of the projetile. The pth tred point on the rim of irle of rdius tht rolls long stright line is lled loid nd is given sin nd os,
5 Chpter Review 833 TABLE Stndrd Equtions for Trnslted Conis rols ( h) 4( k) ( k) 4( h) V Verte ous (h, k ) opens up opens down V Verte ous (h, k) opens left opens right Cirles ( h) ( k) r Rdius r r C ( h) ( k) Ellipses ( h) ( k) Mjor is Minor is Mjor is Minor is ( h) ( k) Hperols ( k) ( h) Trnsverse is Conjugte is Trnsverse is Conjugte is
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