1300 Math Formulas(MATHS 4 ALL)

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