Topics in Analytic Geometry

Transcription

1 Nme Chpter 10 Topics in Anltic Geometr Section 10.1 Lines Objective: In this lesson ou lerned how to find the inclintion of line, the ngle between two lines, nd the distnce between point nd line. Importnt Vocbulr Define ech term or concept. Inclintion (of nonhorizontl line) Angle between two lines I. Inclintion of Line (Pges 78 79) Ever nonhorizontl line must. The ngle formed b such n intersection determines the of the line. How to find the inclintion of line If nonverticl line hs inclintion θ nd slope m, then m =. Emple 1: Find the inclintion of the line given b 1 = + 5. II. The Angle Between Two Lines (Pges ) If two nonperpendiculr lines hve slopes m 1 nd m, the ngle between the two lines is given b tn θ = How to find the ngle between two lines Emple : Find the ngle between the two lines: = nd =

2 170 Chpter 10 Topics in Anltic Geometr III. The Distnce Between Point nd Line (Pges ) The distnce between line nd point not on the line is defined to be... How to find the distnce between point nd line The distnce between the point ( 1, 1 ) nd the line A + B + C = 0 is d =. Emple 3: Find the distnce between the point (1, 1) nd the line = 6 3. Additionl notes Homework Assignment Pge(s) Eercises

3 Section 10. Introduction to Conics: Prbols 171 Nme Section 10. Introduction to Conics: Prbols Objective: In this lesson ou lerned how to write the stndrd form of the eqution of prbol. Importnt Vocbulr Define ech term or concept. Directri Focus Focl chord Ltus rectum Tngent I. Conics (Pge 735) A conic section, or conic, is... Nme the four bsic conic sections: How to recognize conic s the intersection of plne nd doublenpped cone In the formtion of the four bsic conics, the intersecting plne does not pss through the verte of the cone. When the plne does pss through the verte, the resulting figure is (n), such s... II. Prbols (Pges ) A prbol is... How to write equtions of prbols in stndrd form nd grph prbols. The of prbol is the midpoint between the focus nd the directri. The of the prbol is the line pssing through the focus nd the verte.

4 17 Chpter 10 Topics in Anltic Geometr The stndrd form of the eqution of prbol with verticl is hving verte t (h, k) nd directri = k p is The stndrd form of the eqution of prbol with horizontl is hving verte t (h, k) nd directri = h p is The focus lies on the is p units (directed distnce) from the verte. If the verte is t the origin (0, 0), the eqution tkes on one of the following forms: Emple 1: Find the stndrd form of the eqution of the prbol with verte t the origin nd focus (1, 0). III. Applictions of Prbols (Pges ) Describe rel-life sitution in which prbols re used. How to use the reflective propert of prbols to solve rel-life problems The reflective propert of prbol sttes tht the tngent line to prbol t point P mkes equl ngles with the following two lines: 1) ) Homework Assignment Pge(s) Eercises

5 Section 10.3 Ellipses 173 Nme Section 10.3 Ellipses Objective: In this lesson ou lerned how to write the stndrd form of the eqution of n ellipse. Importnt Vocbulr Define ech term or concept. Foci Vertices Mjor is Center Minor is I. Introduction (Pges ) An ellipse is... How to write equtions of ellipses in stndrd form nd grph ellipses. The stndrd form of the eqution of n ellipse centered t (h, k) nd hving horizontl mjor is of length nd minor is of length b, where 0 < b <, is: The stndrd form of the eqution of n ellipse centered t (h, k) nd hving verticl mjor is of length nd minor is of length b, where 0 < b <, is: In both cses, the foci lie on the mjor is, c units from the center, with c =. If the center is t the origin (0, 0), the eqution tkes one of the following forms: or. Emple 1: Sketch the ellipse given b =

6 174 Chpter 10 Topics in Anltic Geometr II. Applictions of Ellipses (Pge 748) Describe rel-life ppliction in which prbols re used. How to use properties of ellipses to model nd solve rel-life problems III. Eccentricit (Pges ) mesures the ovlness of n ellipse. It is given b the rtio e =. For ever ellipse, the vlue of e lies between nd. For n elongted ellipse, the vlue of e is close to. How to find eccentricities of ellipses Additionl notes Homework Assignment Pge(s) Eercises

7 Section 10.4 Hperbols 175 Nme Section 10.4 Hperbols Objective: In this lesson ou lerned how to write the stndrd form of the eqution of hperbol. Importnt Vocbulr Define ech term or concept. Brnches Trnsverse is Conjugte is I. Introduction (Pges ) A hperbol is... How to write equtions of hperbols in stndrd form The line through hperbol s two foci intersects the hperbol t two points clled. The midpoint of hperbol s trnsverse is is the of the hperbol. The stndrd form of the eqution of hperbol centered t (h, k) nd hving horizontl trnsverse is is The stndrd form of the eqution of hperbol centered t (h, k) nd hving verticl trnsverse is is In ech cse, the vertices nd foci re, respectivel, nd c units from the center. Moreover,, b, nd c re relted b the eqution If the center of the hperbol is t the origin (0, 0), the eqution tkes one of the following forms: or.

8 176 Chpter 10 Topics in Anltic Geometr II. Asmptotes of Hperbol (Pges ) The smptotes of hperbol with horizontl trnsverse is re. The smptotes of hperbol with verticl trnsverse is re. Emple 1: Sketch the grph of the hperbol given b 9 = 9. How to find smptotes of nd grph hperbols The eccentricit of hperbol is e =, where the vlues of e re. III. Applictions of Hperbols (Pge 758) Describe rel-life ppliction in which hperbols occur or re used. How to use properties of hperbols to solve rellife problems IV. Generl Equtions of Conics (Pge 759) The grph of A + C + D + E + F = 0 is one of the following: 1) Circle if ) Prbol if 3) Ellipse if 4) Hperbol if How to clssif conics from their generl equtions Emple : Clssif the eqution = 0 s circle, prbol, n ellipse, or hperbol. Homework Assignment Pge(s) Eercises

9 Section 10.5 Rottion of Conics 177 Nme Section 10.5 Rottion of Conics Objective: In this lesson ou lerned how to eliminte the -term in the eqution of conic nd use the discriminnt to identif conic. Importnt Vocbulr Define ech term or concept. Invrint under rottion Discriminnt I. Rottion (Pges ) The generl eqution of conic whose es re rotted so tht the re not prllel to either the -is or the -is contins (n). How to rotte the coordinte es to eliminte the -term in equtions of conics To eliminte this term, ou cn use procedure clled, whose gol is to rotte the - nd -es until the re prllel to the es of the conic. The generl second-degree eqution A + B + C + D + E + F = 0 cn be rewritten s A ( ) + C ( ) + D + E + F = 0 b rotting the coordinte es through n ngle θ, where cot θ =. The coefficients of the new eqution re obtined b mking the substitutions = nd =.

10 178 Chpter 10 Topics in Anltic Geometr II. Invrints Under Rottion (Pges ) The rottion of the coordinte es through n ngle θ tht trnsforms the eqution A + B + C + D + E + F = 0 into the form A ( ) + C ( ) + D + E + F = 0 hs the following rottion invrints: How to use the discriminnt to clssif conics 1) ) 3) The grph of the eqution A + B + C + D + E + F = 0 is, ecept in degenerte cses, determined b its discriminnt s follows: 1) Ellipse or circle if: ) Prbol if : 3) Hperbol if: Emple 1: Clssif the grph of the following conic: = 0 Homework Assignment Pge(s) Eercises

11 Section 10.6 Prmetric Equtions 179 Nme Section 10.6 Prmetric Equtions Objective: In this lesson ou lerned how to rewrite set of prmetric equtions s rectngulr eqution nd find set of prmetric equtions for grph. Importnt Vocbulr Define ech term or concept. Prmeter I. Plne Curves (Pge 771) If f nd g re continuous functions of t on n intervl I, the set of ordered pirs (f(t), g(t)) is (n) C. The equtions = f (t) nd = g(t) re How to evlute sets of prmetric equtions for given vlues of the prmeter for C, nd t is the. II. Sketching Plne Curve (Pge 77) When sketching curve represented b pir of prmetric equtions, ou plot the points in the. Ech set of coordintes (, ) is determined from vlue chosen for the. Plotting the resulting points in the order of incresing vlues of t trces the curve in specific direction, clled the of the curve. How to sketch curves tht re represented b sets of prmetric equtions Emple 1: Sketch the curve described b the prmetric + equtions = t 3 nd = t 1, 1 t 3.

12 180 Chpter 10 Topics in Anltic Geometr III. Eliminting the Prmeter (Pges ) Eliminting the prmeter is the process of... How to rewrite sets of prmetric equtions s single rectngulr equtions b eliminting the prmeter Describe the process used to eliminte the prmeter from set of prmetric equtions. When converting equtions from prmetric to rectngulr form, ou m need to lter... To eliminte the prmeter in equtions involving trigonometric functions, tr using the identities... IV. Finding Prmetric Equtions for Grph (Pges ) Describe how to find set of prmetric equtions for given grph. How to find sets of prmetric equtions for grphs Homework Assignment Pge(s) Eercises

13 Section 10.7 Polr Coordintes 181 Nme Section 10.7 Polr Coordintes Objective: In this lesson ou lerned how to plot points in the polr coordinte sstem nd write equtions in polr form. I. Introduction (Pges ) To form the polr coordinte sstem in the plne, fi point O, clled the or, nd construct from O n initil r clled the. Then ech point P in the plne cn be ssigned s follows: How to plot points on the polr coordinte sstem 1) r = ) θ = In the polr coordinte sstem, points do not hve unique representtion. For instnce, the point (r, θ) cn be represented s or, where n is n integer. Moreover, the pole is represented b (0, θ), where θ is. Emple 1: Plot the point (r, θ) = (, 11/4) on the polr coordinte sstem. / 3/ Emple : Find nother polr representtion of the point (4, /6).

14 18 Chpter 10 Topics in Anltic Geometr II. Coordinte Conversion (Pges ) The polr coordintes (r, θ) re relted to the rectngulr coordintes (, ) s follows... How to convert points from rectngulr to polr form nd vice vers Emple 3: Convert the polr coordintes (3, 3/) to rectngulr coordintes. III. Eqution Conversion (Pge 78) To convert rectngulr eqution to polr form,... How to convert equtions from rectngulr to polr form nd vice vers Emple 4: Find the rectngulr eqution corresponding to the 5 polr eqution r =. sinθ / / / 3/ 3/ 3/ Homework Assignment Pge(s) Eercises

15 Section 10.8 Grphs of Polr Equtions 183 Nme Section 10.8 Grphs of Polr Equtions Objective: In this lesson ou lerned how to grph polr equtions. I. Introduction (Pge 785) Emple 1: Use point plotting to sketch the grph of the polr eqution r = 3cosθ. / How to grph polr equtions b point plotting 3/ II. Smmetr (Pges ) The grph of polr eqution is smmetric with respect to the following if the given substitution ields n equivlent eqution. Substitution 1) The line θ = /: How to use smmetr to sketch grphs of polr equtions ) The polr is: 3) The pole: Emple : Describe the smmetr of the polr eqution r = (1 sinθ ). III. Zeros nd Mimum r-vlues (Pges ) Two dditionl ids to sketching grphs of polr equtions re... How to use zeros nd mimum r-vlues to sketch grphs of polr equtions

16 184 Chpter 10 Topics in Anltic Geometr Emple 3: Describe the zeros nd mimum r-vlues of the polr eqution r = 5cos θ IV. Specil Polr Grphs (Pges ) List the generl equtions tht ield ech of the following tpes of specil polr grphs: How to recognize specil polr grphs Limçons: Rose curves: Circles: Lemnisctes: / / / 3/ / 3/ 3/ / / 3/ Homework Assignment 3/ 3/ Pge(s) Eercises

17 Section 10.9 Polr Equtions of Conics 185 Nme Section 10.9 Polr Equtions of Conics Objective: In this lesson ou lerned how to define conic in terms of eccentricit nd to write equtions of conics in polr form. I. Alterntive Definition of Conics (Pge 793) The locus of point in the plne tht moves so tht its distnce from fied point (focus) is in constnt rtio to its distnce from fied line (directri) is. The constnt rtio is the of the conic nd is denoted b e. Morever, the conic is n ellipse if, prbol if, nd hperbol if. How to define conics in terms of eccentricit For ech tpe of conic, the is t the pole. II. Polr Equtions of Conics (Pges ) The grph of the polr eqution is conic with verticl directri to the right of the pole, where e > 0 is the eccentricit nd p is the distnce between the focus (pole) nd the directri. How to write nd grph equtions of conics in polr form The grph of the polr eqution is conic with verticl directri to the left of the pole, where e > 0 is the eccentricit nd p is the distnce between the focus (pole) nd the directri. The grph of the polr eqution is conic with horizontl directri bove the pole, where e > 0 is the eccentricit nd p is the distnce between the focus (pole) nd the directri.

18 186 Chpter 10 Topics in Anltic Geometr The grph of the polr eqution is conic with horizontl directri below the pole, where e > 0 is the eccentricit nd p is the distnce between the focus (pole) nd the directri. Emple 1: Identif the tpe of conic from the polr eqution 36 r =, nd describe its orienttion sinθ III. Applictions (Pge 796) Describe rel-life ppliction of polr equtions of conics. How to use equtions of conics in polr form to model rel-life problems / / / 3/ / 3/ 3/ / / 3/ Homework Assignment 3/ 3/ Pge(s) Eercises