1300mathformulas

CHAPTER 7. ANALYTIC GEOMETRY 626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí= ó = óM + â(ñ − ñM )I== ïÜÉêÉ=â=áë=íÜÉ=Öê~ÇáÉåíI= m(ñM I óM )=áë=~=éçáåí=çå=íÜÉ=äáåÉK= = = = Figure 101. = 627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë= ó − óN = ñ − ñN == óO − óN ñO − ñN çê= ñ óN ñN óN N = M K= ñO óO N = 141

CHAPTER 7. ANALYTIC GEOMETRY = = = Figure 102. 628. fåíÉêÅÉéí=cçêã= ñ + ó =N= ~Ä = = = Figure 103. = = 142

CHAPTER 7. ANALYTIC GEOMETRY 629. kçêã~ä=cçêã= ñ Åçëβ + ó ëáåβ − é = M = = = = Figure 104. = 630. mçáåí=aáêÉÅíáçå=cçêã= ñ − ñN = ó − óN I== uv ïÜÉêÉ= (uI v)= áë= íÜÉ= ÇáêÉÅíáçå= çÑ= íÜÉ= äáåÉ= ~åÇ= mN(ñNI óN)= äáÉë= çå=íÜÉ=äáåÉK= = 143

CHAPTER 7. ANALYTIC GEOMETRY = = = Figure 105. 631. sÉêíáÅ~ä=iáåÉ= ñ=~= = 632. eçêáòçåí~ä=iáåÉ= ó=Ä= = 633. srê É=Å~ríç+ê=íbÄrèI=ì= ~íáçå=çÑ=~=píê~áÖÜí=iáåÉ= ïÜÉêÉ== l=áë=íÜÉ=çêáÖáå=çÑ=íÜÉ=ÅççêÇáå~íÉëI= u~Ärr ===áááëëë===í~~Üå=âÉóå=é=îçç~ïëêáåáí~á=îÄçÉäåÉÅ==íîéçÉçêÅá=íçåçÑíê==Çç=çáåÑê==É~íÅÜ=âíÉáåç=äçáååïI=ÉéåI=~==éê~çäáäåÉíä==í^ç==çíÜåÉ=í=ÜäáÉå=ÉäáIå==É=I= í=áë=~=é~ê~ãÉíÉêI== rê = → =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~åó=éçáåí=u=çå=íÜÉ=äáåÉK== lu = 144

CHAPTER 7. ANALYTIC GEOMETRY = = Figure 106. = 634. píê~áÖÜí=iáåÉ=áå=m~ê~ãÉíêáÅ=cçêã= ñ = ~N + íÄN I== ó = ~O + íÄO ïÜÉêÉ== (ñI ó) ~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~åó=ìåâåçïå=éçáåí=çå=íÜÉ=äáåÉI== (~NI ~O )=~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=âåçïå=éçáåí=çå=íÜÉ=äáåÉI== (ÄNI ÄO )=~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=îÉÅíçê=é~ê~ääÉä=íç=íÜÉ=äáåÉI== í=áë=~=é~ê~ãÉíÉêK= = 145

CHAPTER 7. ANALYTIC GEOMETRY Figure 107. = = 635. aáëí~åÅÉ=cêçã=~=mçáåí=qç=~=iáåÉ= qÜÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=éçáåí= m(~I Ä) =íç=íÜÉ=äáåÉ= ^ñ + _ó + ` = M =áë== Ç = ^~ + _Ä + ` K= ^O + _O = = = Figure 108. 146

CHAPTER 7. ANALYTIC GEOMETRY 636. m~ê~ääÉä=iáåÉë= qïç=äáåÉë= ó = âNñ + ÄN =~åÇ= ó = âOñ + ÄO =~êÉ=é~ê~ääÉä=áÑ== âN = âO K= qïç= äáåÉë= ^Nñ + _Nó + `N = M = ~åÇ= ^Oñ + _Oó + `O = M = ~êÉ= é~ê~ääÉä=áÑ= ^N = _N K= ^O _O = = = Figure 109. = 637. mÉêéÉåÇáÅìä~ê=iáåÉë= qïç=äáåÉë= ó = âNñ + ÄN =~åÇ= ó = âOñ + ÄO =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ== âO = − N =çêI=Éèìáî~äÉåíäóI= â Nâ O = −N K= âN qïç= äáåÉë= ^Nñ + _Nó + `N = M = ~åÇ= ^Oñ + _Oó + `O = M = ~êÉ= éÉêéÉåÇáÅìä~ê=áÑ= ^N^O + _N_O = M K= = 147

CHAPTER 7. ANALYTIC GEOMETRY = = Figure 110. = 638. ^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë= í~å ϕ = âO − âN I== N+ âNâO Åçëϕ = ^N^O + _N_O K= ^NO + _NO ⋅ ^ O + _OO O = 148

CHAPTER 7. ANALYTIC GEOMETRY = = Figure 111. = 639. fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë= fÑ=íïç=äáåÉë= ^Nñ + _Nó + `N = M =~åÇ= ^Oñ + _Oó + `O = M =áåíÉê- ëÉÅíI=íÜÉ=áåíÉêëÉÅíáçå=éçáåí=Ü~ë=ÅççêÇáå~íÉë= ñM = − `N_O + `O_N I= óM = − ^N`O + ^O`N K= ^N_O − ^O_N ^N_O − ^O_N = = = 7.4 Circle = o~ÇáìëW=o= `ÉåíÉê=çÑ=ÅáêÅäÉW= (~I Ä) = mçáåí=ÅççêÇáå~íÉëW=ñI=óI= ñN I= óN I=£= oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í= 149

CHAPTER 7. ANALYTIC GEOMETRY 640. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ= cçêãF= ñO + óO = oO = ====== = = Figure 112. = 641. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=^åó=mçáåí= (~I Ä) (ñ − ~)O + (ó − Ä)O = oO Figure 113. 150

CHAPTER 7. ANALYTIC GEOMETRY 642. qÜêÉÉ=mçáåí=cçêã ñO + óO ñ ó N ñ O + óNO ñN óN N=M N ñO óO N ñ O + ó O O O ñ O + ó O ñP óP N P P = = = = Figure 114. 643. m~ê~ãÉíêáÅ=cçêã ñ = o Åçë í I= M ≤ í ≤ Oπ K ó = o ëáå í = 644. dÉåÉê~ä=cçêã ^ñO + ^óO + añ + bó + c = M =E^=åçåòÉêçI= aO + bO > Q^c FK== qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÅáêÅäÉ=Ü~ë=ÅççêÇáå~íÉë= (~I Ä) I=ïÜÉêÉ== ~ = − a I= Ä = − b K= O^ O^ qÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅäÉ=áë 151

CHAPTER 7. ANALYTIC GEOMETRY o = aO + bO − Q^c K O^ = = = 7.5 Ellipse = pÉãáã~àçê=~ñáëW=~= pÉãáãáåçê=~ñáëW=Ä= cçÅáW= cN(− ÅI M)I= cO(ÅI M)= aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= = bÅÅÉåíêáÅáíóW=É== oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í= mÉêáãÉíÉêW=i= ^êÉ~W=p= = = 645. bèì~íáçå=çÑ=~å=bääáéëÉ=Epí~åÇ~êÇ=cçêãF ñO + óO =N ~O ÄO = = Figure 115. 152

CHAPTER 7. ANALYTIC GEOMETRY 646. êN + êO = O~ I= ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó==éçáåí== m(ñI ó) ==çå= íÜÉ=ÉääáéëÉ=íç=íÜÉ=íïç=ÑçÅáK= = = = Figure 116. = 647. ~O = ÄO + ÅO = 648. bÅÅÉåíêáÅáíó É= Å <N= ~ = 649. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë ñ = ± ~ = ± ~O = ÉÅ = 650. m~ê~ãÉíêáÅ=cçêã ñ = ~ Åçë í I= M ≤ í ≤ Oπ K ó = Äëáå í = = 153

CHAPTER 7. ANALYTIC GEOMETRY 651. dÉåÉê~ä=cçêã ^ñO + _ñó + `óO + añ + bó + c = M I== ïÜÉêÉ= _O − Q^` < M K= = 652. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë ^ñO + `óO + añ + bó + c = M I== ïÜÉêÉ= ^` > M K = 653. `áêÅìãÑÉêÉåÅÉ i = Q~b(É)I== ïÜÉêÉ==íÜÉ==ÑìåÅíáçå=b==áë==íÜÉ=ÅçãéäÉíÉ==ÉääáéíáÅ=áåíÉÖê~ä==çÑ= íÜÉ=ëÉÅçåÇ=âáåÇK== = 654. ^ééêçñáã~íÉ=cçêãìä~ë=çÑ=íÜÉ=`áêÅìãÑÉêÉåÅÉ ( )i = π NKR(~ + Ä)− ~Ä I== ( )i = π O ~O + ÄO K= = 655. p = π~Ä = = = = 7.6 Hyperbola = qê~åëîÉêëÉ=~ñáëW=~= `çåàìÖ~íÉ=~ñáëW=Ä= cçÅáW= cN(− ÅI M)I= cO(ÅI M)= aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= = bÅÅÉåíêáÅáíóW=É== ^ëóãéíçíÉëW=ëI=í= oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=íI=â= = = = 154

CHAPTER 7. ANALYTIC GEOMETRY 656. bèì~íáçå=çÑ=~=eóéÉêÄçä~=Epí~åÇ~êÇ=cçêãF= ñO − óO = N= ~O ÄO = = = Figure 117. = 657. êN − êO = O~ I= ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó=éçáåí== m(ñIó) ==çå= íÜÉ=ÜóéÉêÄçä~=íç=íÜÉ=íïç=ÑçÅáK= = 155

CHAPTER 7. ANALYTIC GEOMETRY = = Figure 118. = 658. bèì~íáçåë=çÑ=^ëóãéíçíÉë= ó=±Äñ= ~ = 659. ÅO = ~O + ÄO = = 660. bÅÅÉåíêáÅáíó É = Å >N= ~ = 661. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë ñ = ± ~ = ± ~O = ÉÅ = = = 156

CHAPTER 7. ANALYTIC GEOMETRY 662. m~ê~ãÉíêáÅ=bèì~íáçåë=çÑ=íÜÉ=oáÖÜí=_ê~åÅÜ=çÑ=~=eóéÉêÄçä~= ñ = ~ ÅçëÜ í I= M ≤ í ≤ Oπ K ó = Ä ëáåÜ í = 663. dÉåÉê~ä=cçêã ^ñO + _ñó + `óO + añ + bó + c = M I== ïÜÉêÉ= _O − Q^` > M K= = 664. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë ^ñO + `óO + añ + bó + c = M I== ïÜÉêÉ= ^` < M K= 665. ^ëóãéíçíáÅ=cçêã= ñó = ÉO I== Q çê== ó = â I=ïÜÉêÉ= â = ÉO K= ñQ få= íÜáë= Å~ëÉ= I= íÜÉ= ~ëóãéíçíÉë= Ü~îÉ= Éèì~íáçåë= ñ = M = ~åÇ= ó = M K== = 157

CHAPTER 7. ANALYTIC GEOMETRY = = = Figure 119. = = 7.7 Parabola = cçÅ~ä=é~ê~ãÉíÉêW=é= cçÅìëW=c= sÉêíÉñW= j(ñM I óM ) = oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=éI=~I=ÄI=Å= = = 666. bèì~íáçå=çÑ=~=m~ê~Äçä~=Epí~åÇ~êÇ=cçêãF óO = Oéñ = 158

CHAPTER 7. ANALYTIC GEOMETRY = = Figure 120. = bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ ñ = − é I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë= c é I M I=  O  `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= j(MI M) K= = 667. dÉåÉê~ä=cçêã ^ñO + _ñó + `óO + añ + bó + c = M I== ïÜÉêÉ= _O − Q^` = M K= = 668. ó = ~ñO I= é = N K= O~ bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ 159

CHAPTER 7. ANALYTIC GEOMETRY ó = − é I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë= c MI é  I=  O  `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= j(MI M) K= = = = Figure 121. = 669. dÉåÉê~ä=cçêãI=^ñáë=m~ê~ääÉä=íç=íÜÉ=ó-~ñáë== ^ñO + añ + bó + c = M =E^I=b=åçåòÉêçFI== ó = ~ñO + Äñ + Å I= é = N K== O~ bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ ó = ó M − é I= O `ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë= 160

CHAPTER 7. ANALYTIC GEOMETRY c ñM I óM + é  I=  O  `ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ= ñM = − Ä I= óM = ~ñ O + Äñ M + Å = Q~Å − ÄO K= O~ M Q~ = = = Figure 122. = = = 7.8 Three-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñM I= óM I= òM I= ñN I= óN I= òN I=£= oÉ~ä=åìãÄÉêW= λ == aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= ^êÉ~W=p= sçäìãÉW=s= = 161

CHAPTER 7. ANALYTIC GEOMETRY 670. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= Ç = ^_ = (ñO − ñN )O + (óO − óN )O + (òO − òN )O = = === = = Figure 123. = 671. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñM = ñN + λñ O I= óM = óN + λóO I= òM = òN + λòO I== N+ λ N+ λ N+ λ ïÜÉêÉ= λ = ^` I= λ ≠ −NK= `_ = 162

CHAPTER 7. ANALYTIC GEOMETRY ======== = = Figure 124. = = Figure 125. 163

CHAPTER 7. ANALYTIC GEOMETRY 672. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= ñM = ñN + ñO I= óM = óN + óO I= òM = òN + òO I= λ = N K= O O O = 673. ^êÉ~=çÑ=~=qêá~åÖäÉ= qÜÉ=~êÉ~=çÑ=~=íêá~åÖäÉ=ïáíÜ=îÉêíáÅÉë= mN(ñNI óNIòN) I= mO (ñO I óO IòO )I=~åÇ= mP (ñP I óP IòP )=áë=ÖáîÉå=Äó== p= N óN òN NO òN ñN NO ñN óN NO O óO òO N + òO ñO N + ñO óO N K= óP òP N òP ñP N ñP óP N = 674. sçäìãÉ=çÑ=~=qÉíê~ÜÉÇêçå= qÜÉ=îçäìãÉ=çÑ=~=íÉíê~ÜÉÇêçå=ïáíÜ=îÉêíáÅÉë= mN(ñNI óNIòN) I= mO (ñO I óO IòO )I= mP (ñP I óP IòP )I=~åÇ= mQ (ñQ I óQ IòQ )=áë=ÖáîÉå=Äó== ñN óN òN N s = ± N ñO óO òO N I== S ñP óP òP N ñQ óQ òQ N çê= s = ± N ñN − ñQ óN − óQ òN − òQ S ñO − ñQ óO − óQ òO − òQ K= ñP − ñQ óP − óQ òP − òQ kçíÉW=tÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç=íÜ~í=íç=ÖÉí=~=éçëáíáîÉ= ~åëïÉê=Ñçê=îçäìãÉK== = 164

CHAPTER 7. ANALYTIC GEOMETRY ==== = = Figure 126. = = = 7.9 Plane = mçáåí=ÅççêÇáå~íÉëW=ñI=óI=òI= ñM I= óM I= òM I= ñN I= óN I= òN I=£= okÉç~êäã=å~ìäã=îÉÄÅÉíêçëêW=ë^W=Iår=_II==å`rNII=a= årIO= ^N I= ^ O I=~I=ÄI=ÅI= ~N I= ~ O I= λ I=éI=íI=£== = aáêÉÅíáçå=ÅçëáåÉëW= Åçë α I= Åçëβ I= Åçë γ = aáëí~åÅÉ=Ñêçã=éçáåí=íç=éä~åÉW=Ç= = = 675. dÉåÉê~ä=bèì~íáçå=çÑ=~=mä~åÉ= ^ñ + _ó + `ò + a = M = = = 165

CHAPTER 7. ANALYTIC GEOMETRY 676. qkÜçÉê=ãîÉ~Åäí=sçêÉ=Ååríç(ê^=íIç_=~I `=m)ä~=áåë=Éå=çêã~ä=íç=íÜÉ=éä~åÉ= ^ñ + _ó + `ò + a = M K= = === = = Figure 127. = 677. m~êíáÅìä~ê=`~ëÉë=çÑ=íÜÉ=bèì~íáçå=çÑ=~=mä~åÉ= ^ñ + _ó + `ò + a = M = = fÑ= ^ = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ñ-~ñáëK= fÑ= _ = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ó-~ñáëK= fÑ= ` = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ò-~ñáëK= fÑ= a = M I=íÜÉ=éä~åÉ=äáÉë=çå=íÜÉ=çêáÖáåK== = fÑ= ^ = _ = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ñó-éä~åÉK= fÑ= _ = ` = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=óò-éä~åÉK= fÑ= ^ = ` = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ñò-éä~åÉK= = 166

CHAPTER 7. ANALYTIC GEOMETRY 678. mçáåí=aáêÉÅíáçå=cçêã= ^(ñ − ñM )+ _(ó − óM )+ `(ò − òM ) = M I== ïÜÉêÉ=íÜÉ=éçáåí= m(ñM I óM IòM )=äáÉë=áå=íÜÉ=éä~åÉI=~åÇ=íÜÉ=îÉÅ- íçê= (^I _I `)=áë=åçêã~ä=íç=íÜÉ=éä~åÉK=== = ==== = = Figure 128. = 679. fåíÉêÅÉéí=cçêã= ñ + ó + ò = N= ~ Ä Å = 167

CHAPTER 7. ANALYTIC GEOMETRY ===== = = Figure 129. = 680. qÜêÉÉ=mçáåí=cçêã= ñ − ñP ó − óP ò −òP ñN − ñP óN − óP òN − òP = M I== ñO − ñP óO − óP òO − òP çê== ñ ó òN ñN óN òN N = M K= ñO óO òO N ñP óP òP N = 168

CHAPTER 7. ANALYTIC GEOMETRY ===== = = Figure 130. = 681. kçêã~ä=cçêã= ñ Åçëα + ó Åçëβ + ò Åçë γ − é = M I== ïÜÉêÉ==é==áë==íÜÉ==éÉêéÉåÇáÅìä~ê==Çáëí~åÅÉ==Ñêçã==íÜÉ=çêáÖáå=íç= íÜÉ=éä~åÉ=I=~åÇ= Åçë α I= Åçëβ I= Åçë γ =~êÉ=íÜÉ=ÇáêÉÅíáçå=ÅçëáåÉë= çÑ=~åó=äáåÉ=åçêã~ä=íç=íÜÉ=éä~åÉK== = 169

CHAPTER 7. ANALYTIC GEOMETRY ====== = = Figure 131. = 682. m~ê~ãÉíêáÅ=cçêã= ñ = ñN + ~Në + ~Oí I== ó = óN + ÄNë + ÄOí ò = òN + ÅNë + ÅOí ïÜÉêÉ= (ñI óIò) =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~åó=ìåâåçïå=éçáåí=çå= íÜÉ=äáåÉ=I=íÜÉ=éçáåí= m(ñNIóNIòN)=äáÉë=áå=íÜÉ=éä~åÉI=íÜÉ=îÉÅíçêë= (~NIÄNIÅN) =~åÇ= (~O IÄO IÅO ) =~êÉ=é~ê~ääÉä=íç=íÜÉ=éä~åÉK= = 170

CHAPTER 7. ANALYTIC GEOMETRY ===== = = Figure 132. = 683. aáÜÉÇê~ä=^åÖäÉ=_ÉíïÉÉå=qïç=mä~åÉë= fÑ=íÜÉ=éä~åÉë=~êÉ=ÖáîÉå=Äó== ^Nñ + _Nó + `Nò + aN = M I== ^Oñ + _Oó + `Oò + aO = M I== íÅÜçÉëåϕ=í=ÜÉårå=rÇNNá⋅⋅ÜårårÉOÇO ê=~ä=~å^ÖNOäÉ+^=Ä_NÉ^NOíOï++É`É_NOåN⋅=_íÜO^É+ãOO`+=áNë`_==OOO + ` O K= O = 171

CHAPTER 7. ANALYTIC GEOMETRY ====== = = Figure 133. = 684. m~ê~ääÉä=mä~åÉë= qïç=éä~åÉë= ^Nñ + _Nó + `Nò + aN = M =~åÇ= ^Oñ + _Oó + `Oò + aO = M =~êÉ=é~ê~ääÉä=áÑ== ^N = _N = `N K= ^O _O `O = 685. mÉêéÉåÇáÅìä~ê=mä~åÉë= qïç=éä~åÉë= ^Nñ + _Nó + `Nò + aN = M =~åÇ= ^Oñ + _Oó + `Oò + aO = M =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ== ^N^O + _N_O + `N`O = M K= = 686. bèì~íáçå=çÑ=~=mä~åÉ=qÜêçìÖÜ= m(ñNI óNIòN )=~åÇ=m~ê~ääÉä=qç= íÜÉ=sÉÅíçêë= (~NIÄNIÅN) =~åÇ= (~O IÄO IÅO ) =EcáÖKNPOF= 172

CHAPTER 7. ANALYTIC GEOMETRY ñ − ñN ó − óN ò − òN ~N ÄN ÅN = M = ~O ÄO ÅO = 687. bèì~íáçå=çÑ=~=mä~åÉ=qÜêçìÖÜ= mN(ñNI óNIòN) =~åÇ= mO(ñO I óO IòO )I= ~åÇ=m~ê~ääÉä=qç=íÜÉ=sÉÅíçê= (~I ÄI Å) = ñ − ñN ó − óN ò − òN ñO − ñN óO − óN òO − òN = M = ~ Ä Å = = Figure 134. = 688. aáëí~åÅÉ=cêçã=~=mçáåí=qç=~=mä~åÉ= qÜÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=éçáåí= mN(ñNI óNIòN) =íç=íÜÉ=éä~åÉ= ^ñ + _ó + `ò + a = M =áë== 173

CHAPTER 7. ANALYTIC GEOMETRY Ç = ^ñN + _óN + `òN + a K= ^O + _O + `O = ====== = = Figure 135. = 689. fåíÉêëÉÅíáçå=çÑ=qïç=mä~åÉë= fÑ=íïç=éä~åÉë= ^Nñ + _Nó + `Nò + aN = M =~åÇ= ^Oñ + _Oó + `Oò + aO = M = áåíÉêëÉÅíI= íÜÉ= áåíÉêëÉÅíáçå= ëíê~áÖÜí= äáåÉ=áë=ÖáîÉå=Äó= ñ = ñN + ~í I== ó = óN + Äí ò = òN + Åí çê== ñ − ñN = ó − óN = ò − òN I== ~ ÄÅ ïÜÉêÉ== 174

CHAPTER 7. ANALYTIC GEOMETRY ~ = _N `N I= Ä = `N ^N I= Å = ^N _N I== _O `O `O ^O ^O _O Ä aN `N − Å aN _N aO `O aO _O I== ñN = ~O + ÄO + ÅO Å aN ^N − ~ aN `N aO ^O aO `O I== óN = ~O + ÄO + ÅO ~ aN _N − Ä aN ^N aO _O aO ^O K== òN = ~O + ÄO + ÅO = = = 7.10 Straight Line in Space = mçáåí=ÅççêÇáå~íÉëW=ñI=óI=òI= ñN I= óN I= òN I=£= aáêÉÅíáçå=ÅçëáåÉëW= Åçë α I= Åçëβ I= Åçë γ = oakÉçáê~êÉäã=Ååí~ìáçäã=åîÉ=ÄîÅÉÉíêÅçëíêWç==^íêçëI===_~ç=ÑIé==`~ä~=Iäå=áaåÉÉWI==W~å=rIrë==ÄI=I=rëÅNII==~rëNOI== ~O I=íI=£== ^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW= ϕ = = 690. mçáåí=aáêÉÅíáçå=cçêã=çÑ=íÜÉ=bèì~íáçå=çÑ=~=iáåÉ== ñ − ñN = ó − óN = ò − òN I== ~ ÄÅ ïÜÉêÉ=íÜÉ=éçáåí= mN(ñNI óNIòN) =äáÉë=çå=íÜÉ=äáåÉI=~åÇ= (~IÄIÅ) =áë= íÜÉ=ÇáêÉÅíáçå=îÉÅíçê=çÑ=íÜÉ=äáåÉK== = 175

CHAPTER 7. ANALYTIC GEOMETRY ===== = = Figure 136. = 691. qïç=mçáåí=cçêã= ñ − ñN = ó − óN = ò − òN == ñO − ñN óO − óN òO − òN = ===== = Figure 137. 176

CHAPTER 7. ANALYTIC GEOMETRY 692. m~ê~ãÉíêáÅ=cçêã== ñ = ñN + í Åçë α I== ó = óN + í Åçëβ ò = òN + í Åçë γ ïÜÉêÉ=íÜÉ=éçáåí= mN(ñNI óNIòN) =äáÉë=çå=íÜÉ=ëíê~áÖÜí=äáåÉI= Åçë α I= Åçëβ I= Åçë γ =~êÉ=íÜÉ=ÇáêÉÅíáçå=ÅçëáåÉë=çÑ=íÜÉ=ÇáêÉÅíáçå= îÉÅíçê=çÑ=íÜÉ=äáåÉI=íÜÉ=é~ê~ãÉíÉê=í=áë=~åó=êÉ~ä=åìãÄÉêK== = ===== = = Figure 138. = 693. ^ÅçåëÖϕäÉ==_ÉrërëNíNï⋅⋅ rëÉrëOÉO å==qï~çNO=p+í~êÄN~~NOáOÖ++ÜÅíÄNO=iN⋅ÄáåO ~É+ëOO=Å+NÅÄOOO + Å O = O = 177

CHAPTER 7. ANALYTIC GEOMETRY ===== = = Figure 139. = 694. m~ê~ääÉä=iáåÉë= qrëNïööçrë=OäáIå==Éë=~êÉ=é~ê~ääÉä=áÑ== çê== ~N = ÄN = ÅN K= ~O ÄO ÅO = 695. mÉêéÉåÇáÅìä~ê=iáåÉë= qrëNï⋅ rëçO=ä=áåMÉëI===~êÉ=é~ê~ääÉä=áÑ== çê== ~N~O + ÄNÄO + ÅNÅO = M K= = 696. fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë= qïç=äáåÉë= ñ − ñN = ó − óN = ò − òN =~åÇ= ~N ÄN ÅN 178

CHAPTER 7. ANALYTIC GEOMETRY ñ − ñO = ó − óO = ò − òO =áåíÉêëÉÅí=áÑ== ~O ÄO ÅO ñO − ñN óO − óN òO − òN ~N ÄN ÅN = M K= ~O ÄO ÅO = 697. m~ê~ääÉä=iáåÉ=~åÇ=mä~åÉ== qÜÉ=ëíê~áÖÜí=äáåÉ= ñ − ñN = ó − óN = ò − òN =~åÇ=íÜÉ=éä~åÉ= ~ ÄÅ å^r ⋅ñrë+=_MóI=+= `ò + a = M =~êÉ=é~ê~ääÉä=áÑ= çê== ^~ + _Ä + `Å = M K= = ===== = = Figure 140. = = 179

CHAPTER 7. ANALYTIC GEOMETRY 698. mÉêéÉåÇáÅìä~ê=iáåÉ=~åÇ=mä~åÉ== qÜÉ=ëíê~áÖÜí=äáåÉ= ñ − ñN = ó − óN = ò − òN =~åÇ=íÜÉ=éä~åÉ= ~ ÄÅ å^r öñö rë+I_==ó + `ò + a = M =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ= çê== ^ = _ = ` K= ~ÄÅ = ==== = = Figure 141. = = 7.11 Quadric Surfaces = mçáåí=ÅççêÇáå~íÉë=çÑ=íÜÉ=èì~ÇêáÅ=ëìêÑ~ÅÉëW=ñI=óI=ò= oÉ~ä=åìãÄÉêëW=^I=_I=`I=~I=ÄI=ÅI= âNIâO IâP I=£= 180

CHAPTER 7. ANALYTIC GEOMETRY 699. dÉåÉê~ä=nì~Çê~íáÅ=bèì~íáçå= ^ñO + _óO + `òO + Ocóò + Odòñ + Oeñó + Omñ + Onó + Ooò + a = M = 700. `ä~ëëáÑáÅ~íáçå=çÑ=nì~ÇêáÅ=pìêÑ~ÅÉë= = `~ëÉ= o~åâEÉF= o~åâEbF= ∆ = â=ëáÖåë= qóéÉ=çÑ=pìêÑ~ÅÉ= Q= < M = p~ãÉ= oÉ~ä=bääáéëçáÇ= N= P= O= P= Q= > M = p~ãÉ= fã~Öáå~êó=bääáéëçáÇ= P= P= Q= > M = aáÑÑÉêÉåí= eóéÉêÄçäçáÇ=çÑ=N=pÜÉÉí= Q= P= Q= < M = aáÑÑÉêÉåí= eóéÉêÄçäçáÇ=çÑ=O=pÜÉÉíë= R= P= P= = aáÑÑÉêÉåí= oÉ~ä=nì~ÇêáÅ=`çåÉ= S= P= P= = p~ãÉ= fã~Öáå~êó=nì~ÇêáÅ=`çåÉ= T= O= bääáéíáÅ=m~ê~ÄçäçáÇ= Q= < M = p~ãÉ= U= O= Q= > M = aáÑÑÉêÉåí= eóéÉêÄçäáÅ=m~ê~ÄçäçáÇ= V= O= P= = p~ãÉ= oÉ~ä=bääáéíáÅ=`óäáåÇÉê= NM= O= P= = p~ãÉ= fã~Öáå~êó=bääáéíáÅ=`óäáåÇÉê= NN= O= P= = aáÑÑÉêÉåí= eóéÉêÄçäáÅ=`óäáåÇÉê= NO= O= O= = aáÑÑÉêÉåí= oÉ~ä=fåíÉêëÉÅíáåÖ=mä~åÉë= NP= O= O= = p~ãÉ= fã~Öáå~êó=fåíÉêëÉÅíáåÖ=mä~åÉë= NQ= N= P= = = m~ê~ÄçäáÅ=`óäáåÇÉê= NR= N= O= = = oÉ~ä=m~ê~ääÉä=mä~åÉë= NS= N= O= = = fã~Öáå~êó=m~ê~ääÉä=mä~åÉë= NT= N= N= = = `çáåÅáÇÉåí=mä~åÉë= = eÉêÉ==  ^ e d  ^ e n m  É = e _ c  e _ c n c `  I= b =  d c ` ao  I= ∆ = ÇÉí(b)I==  d  m n o âNIâO IâP =~êÉ=íÜÉ=êççíë=çÑ=íÜÉ=Éèì~íáçåI== ^−ñ e d e _ − ñ c = M K= d c `−ñ = = 181

CHAPTER 7. ANALYTIC GEOMETRY 701. oÉ~ä=bääáéëçáÇ=E`~ëÉ=NF= ñO + óO + òO = N= ~O ÄO ÅO = ===== = = Figure 142. = 702. fã~Öáå~êó=bääáéëçáÇ=E`~ëÉ=OF= ñO + óO + òO = −N= ~O ÄO ÅO = 703. eóéÉêÄçäçáÇ=çÑ=N=pÜÉÉí=E`~ëÉ=PF= ñO + óO − òO =N= ~O ÄO ÅO = 182

CHAPTER 7. ANALYTIC GEOMETRY ===== = = Figure 143. = 704. eóéÉêÄçäçáÇ=çÑ=O=pÜÉÉíë=E`~ëÉ=QF= ñO + óO − òO = −N = ~O ÄO ÅO = = = = Figure 144. 183

CHAPTER 7. ANALYTIC GEOMETRY 705. oÉ~ä=nì~ÇêáÅ=`çåÉ=E`~ëÉ=RF= ñO + óO − òO =M= ~O ÄO ÅO = == = = Figure 145. = 706. fã~Öáå~êó=nì~ÇêáÅ=`çåÉ=E`~ëÉ=SF= ñO + óO + òO =M= ~O ÄO ÅO = 707. bääáéíáÅ=m~ê~ÄçäçáÇ=E`~ëÉ=TF= ñO + óO −ò = M = ~O ÄO = 184

CHAPTER 7. ANALYTIC GEOMETRY ====== = Figure 146. 708. eóéÉêÄçäáÅ=m~ê~ÄçäçáÇ=E`~ëÉ=UF= ñO − óO −ò = M = ~O ÄO ==== = Figure 147. 185

CHAPTER 7. ANALYTIC GEOMETRY 709. oÉ~ä=bääáéíáÅ=`óäáåÇÉê=E`~ëÉ=VF= ñO + óO = N= ~O ÄO = ===== = = Figure 148. = 710. fã~Öáå~êó=bääáéíáÅ=`óäáåÇÉê=E`~ëÉ=NMF= ñO + óO = −N= ~O ÄO = 711. eóéÉêÄçäáÅ=`óäáåÇÉê=E`~ëÉ=NNF= ñO − óO = N= ~O ÄO = 186

CHAPTER 7. ANALYTIC GEOMETRY ====== = = Figure 149. = 712. oÉ~ä=fåíÉêëÉÅíáåÖ=mä~åÉë=E`~ëÉ=NOF= ñO − óO =M= ~O ÄO = 713. fã~Öáå~êó=fåíÉêëÉÅíáåÖ=mä~åÉë=E`~ëÉ=NPF= ñO + óO =M= ~O ÄO = 714. m~ê~ÄçäáÅ=`óäáåÇÉê=E`~ëÉ=NQF= ñO − ó = M= ~O = 187

CHAPTER 7. ANALYTIC GEOMETRY ===== = = Figure 150. = 715. oÉ~ä=m~ê~ääÉä=mä~åÉë=E`~ëÉ=NRF= ñO = N= ~O = 716. fã~Öáå~êó=m~ê~ääÉä=mä~åÉë=E`~ëÉ=NSF= ñO = −N = ~O = 717. `çáåÅáÇÉåí=mä~åÉë=E`~ëÉ=NTF= ñO = M= = = = = = 188

CHAPTER 7. ANALYTIC GEOMETRY 7.12 Sphere = o~Çáìë=çÑ=~=ëéÜÉêÉW=o= mçáåí=ÅççêÇáå~íÉëW=ñI=óI=òI= ñN I= óN I= òN I=£= `ÉåíÉê=çÑ=~=ëéÜÉêÉW= (~IÄIÅ) = oÉ~ä=åìãÄÉêëW=^I=aI=bI=cI=j= = = 718. bèì~íáçå=çÑ=~=péÜÉêÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ= cçêãF= ñO + óO + òO = oO = = ====== = = Figure 151. = 719. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=^åó=mçáåí= (~IÄIÅ) (ñ − ~)O + (ó − Ä)O + (ò − Å)O = oO 720. aá~ãÉíÉê=cçêã (ñ − ñN)(ñ − ñO )+ (ó − óN)(ó − óO )+ (ò − òN)(ò − òO ) = M I== 189

CHAPTER 7. ANALYTIC GEOMETRY ïÜÉêÉ== mN(ñNI óNIòN ) I= mO (ñO I óO IòO )=~êÉ=íÜÉ=ÉåÇë=çÑ=~=Çá~ãÉíÉêK== = 721. cçìê=mçáåí=cçêã= ñO + óO +òO ñ ó ò N ñ O + óNO + ñ NO ñN óN òN N N ñ O + ó O + ñ O ñO óO òO N =M= O O O ñ O + ó PO + ñ O ñP óP òP N P P ñ O + ó O + ñ O ñQ óQ òQ N Q Q Q = 722. dÉåÉê~ä=cçêã ^ñO + ^óO + ^òO + añ + bó + cò + j = M =E^=áë=åçåòÉêçFK== qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ëéÜÉêÉ=Ü~ë=ÅççêÇáå~íÉë= (~IÄIÅ) I=ïÜÉêÉ== ~ = − a I= Ä = − b I= Å = − c K= O^ O^ O^ qÜÉ=ê~Çáìë=çÑ=íÜÉ=ëéÜÉêÉ=áë o = aO + bO + cO − Q^Oj K O^ = = 190