Agma Notes

8/12/2019 Agma Notes

1/106


2/106


3/106

u ,v , . . .

u A B


4/106

AB A

AA

u =AB u AB u |u |

|u | = |AB|=

A

B

u

u

u

u ,v u //v

AB, BC, CD

DA M AC

BD ABCD


5/106

ABCD

BCDA

CDAB

DABC

ADCB

DCBA

CBAD

BADC AC BD

ABCD ABDC

A= B C=D

ABCD

CDEF

ABFE


6/106

AB AB

u = AB A

AB

AB

A

B ABB A

A

AB ABBA

AB

AB

CD AB


7/106

AB

CD

AB CD

u =AB v =CD

CD

AD

D

A

B

u v

D A B

u v


8/106

O

O

u = OA v = OB O u + v O C OACB

u = OA v = OB u +v

OC C |AC| = |OB| |BC| =

|OA|.

u + u

u

u =OA a aR a u

O |a| |u |

u |a|

a

au

u a >0 a


9/106

u + OO = u

1 u = u 0 u = OO

(ab) u =a (b u )

(a + b) u =a u + b u

a (u + v) =a u + a v

u = OA

v = OB

w = OC

D E Z1 Z2

OA+

OB =

OD

OD+

OC=

OZ1

OB+OC=

OE

OA+

OE=

OZ2

(u + v) + w = OZ1, u + (v + w ) = OZ2.


10/106

Z2 OAZ2E

OZ1=

OZ2 OAZ1E

OZ1

AE ODZ1C Z1

OZ1 CD

CD AE ACED

OCEB OBDA E D

ACED

(1)u u + (1)u =OO

u

u v u = v + (u )

AB AB AC AC

AB+

AC AB+ AC

a R,a AB a AB.


11/106

ABB A

ACCA ABDC ABDC ADDA

ABDC ACCA BDCA

BDCA

ABDC

BDDB

BDDB ABB A ADDA

u = OA v = OB a R

v =a u

a v u : a= |OB||OA| . u v v =a u u v v = a u .

u v v =a u

u +v = (1+ a) u

u v

u v

u =OA v =OB

w O,w = OC

a, b R

w =au + bv .


12/106

u v

OC

C

w

OB

u ,v OA A

OA u OA =a u

a R C OA OB B

OB = b v b R

OACB

w = OC = OA+ OB = au + bv .


13/106

u 1,u 2, . . . ,u n O

u 1,u 2, . . . ,u n

a1u 1+ a2u 2+ . . .+ anu n

a1, a2, . . . , an

u 1,u 2, . . . ,u n

u 1,u 2, . . . ,u n

u 1,u 2, . . . ,u n O

i, 1 i n a1, . . . , ai1, ai+1, . . . , an

u i = aiu 1+ . . .+ ai1u i1+ ai+1u i+1+ . . .+ anu n

u 1,u 2, . . . ,u n

O

O

v = OA v

u 1


14/106

u =OB v

u (, v)

a R

u = a|v |v

|a| =|u |

u

a > 0

u

v a < 0 u v

OB (

OB)

(u ) u

Chasles A, B, C

(,v) (AB) + (

BC) = (

AC) .

u =OA v =OB v A A u v

prv (u )

OA

prv(u + w ) = prvu + prvw

prv(a u ) =a prvu prvu =OA prvw =

OB prv(u +w ) =

OC

B

C

D

OAA

BC D (OA) = (BD) = (BC) Chasles

(OC) = (OB )+(BC (OC) = (OB )+(OA).

OC =OA+

OB .

OA

(, 1

|v | v)


15/106

pr

vu =OA u v pr

u

v =OB v u


16/106

u ,v O

|u |(pru v) = |v |(prvu ) .

OAA OBB

|prvu | = 0

|u ||v | =

|OA||OB|

=|OA||OB |

=|prvu ||pru v |

(prvu ) prvu v

u

v

O v u v O u pru v u (pru v)

|v | (prvu ) u

v u v = |v | (prvu ).

u ,v u v = 0.

u ,v ,w

O

a R

u v = v u

(a u ) v =a (u v)

u (v + w ) = u v + u w

u u 0 |u | = u u


17/106

pru u = u

u

u u u = |u | (pruu ) = |u | |u | = |u |2 0.

u = OA v = OB

AOB = 0 (u ,v)

(u ,v)

<

u v (u ,v) = (v , u ) = (u ,v)

(prvu ) = |u | cos(u ,v)

u v = |u ||v | cos(u ,v) .


18/106

E2 O

O

u =OA

v =OB

u

v (,u ) (,v) O (,u ), (,v) (O,u ,v) O

w O

w

w =a u + b v .

(a, b)

w (O,u ,v) a u , b v

w (O,u ,v) C C


19/106

OC

OC

C

(O,u ,v)

E2

R2 = {(x, y)|x, y R}

u ,v i =

1

|u |u , j = 1|v |

v .

(O,

i ,j )

(a, b) w =a i + b j


20/106

w

a= w

i , b= w

j

i i =j j = 1 i j = 0

w i = (a i + b j ) i= a

i i + b j i

= a

w j =b .

(O,

i ,

j) u ,v O (u1, u2), (v1, v2) u =u1 i +u2 j v =v1 i +v2 j u +v (u1+ v1, u2+ v2)

u + v = (u1+ v1) i + (u2+ v2) j ,

a R a u (au1, au2)

a u =au1 i + au2 j .


21/106

u v u

v = (u1i + u2j )

(v1

i + v2

j)

= u1v1i i + u1v2i j + u2v1j i + u2v2j j= u1v1+ u2v2.

u

|u | =u u

=

(u1

i + u2

j ) (u1i + u2j)

= u12i i + u1u2i j + u2u1j i + u22j j

=

u12 + u22.

O

AB A


22/106

B B C

AB

AC

AB+ BC

A B

v A A

B

u

A

B v u

F1 A F2 B

F1+

F2.

u = AB

u u

u [u ]


23/106

[u ] = {v v u }. u

v u v

[u ] = [v] u v .

[u ] = [v] [v] = [w ] [u ] = [w ]

12

, 24

, 510

, 341682

12

13

1

2

+ 1

3

= 3

6

+ 2

6

= 5

6

[u ]

[v]

[u ] [v] O OAu

OBv OA+ OB =OC

[u ] [v]

OC

[u ] + [v] =OC

= [w ] .

[u ]

a

aOA

a [u ] =

aOA

= [a u ] .

pr[v][u ] =

prOBOA


24/106

[u ] [v] = OA OB .

OA

OB

OA

OB

OA OA OB OB OA +OB OA +OB [u ] + [v]

[ ] u AB

(O,i ,j)

A, B, C, D

(a1, a2), (b1, b2), (c1, c2) (d1, d2) AB CD ABM CDN

b1 a1 = d1 c1 b2 a2=d2 c2

v1 = b1 a1 v2 = b2 a2

AB v = AB

v (O,

i ,

j)

v1=b1 a1 v2=b2 a2,

(a1, a2) (b1, b2) AB

v

v1= 0 v =

v2v1


25/106

P1, P2, P P1P P P2 P=P2

P1P =

P P2.

(P1 P2 P)

(P1 P2 P) = (

P1P)

(P P2)

P1, P2 P

P1(x1, y1), P2(x2, y2) P(x, y)

x x1=(x2 x) y y1 = (y2 y),

x=x1+ x2

1 + y=

y1+ y21 +

.

>0 P P1 P2

P1P2 : 1


26/106

P

P1P2

ABC (a1, a2)

(b1, b2) (c1, c2)

A(a1, a2) BC


27/106

a1 =b1+ c1

2 a2=

b2+ c22

.

B(b1, b2)

b1 =a1+ c1

2 b2=

a2+ c22

.

AA BB

OA + AA

OB+

mBB m

OA+

AA=

OB+ m

BB .

a1+

b1+ c1

2 a1

= b1+ m

a1+ c1

2 b1

a2+

b2+ c2

2 a2

= b2+ m

a2+ c2

2 b2

, m

a1+b1

2

+c1

2 + b1

a1

2c1

2m= b1 a1

a2+ b22

+c2

2

+

b2 a2

2 c2

2

m= b2 a2

AA,

BB

= 23

, m= 23

.

AA BB G

(AAG) = (BB G) = 2 G AA CC

(AAG) = (CCG) = 2,

G= G

G

OG =

OA+

2

3

AA

= OA+23

(OA OA)


28/106

= 1

3

OA+

2

3(

1

2(OB+

OC))

= 1

3(OA+

OB+

OC)

=

a1+ b1+ c1

3 ,

a2+ b2+ c23

.

(O,

i ,

j)

(P,u ,v)

(P,u ,v) (O,

i ,

j)

A (x, y) (O,

i ,

j)

(x, y) (P,u ,v) OA = x

i + y

j

P A= xu + yv

P A= OA OP


29/106

P (O,

i ,

j) (p, q)

x

u + y

v = (x

p)

i + (y

q)

j

i ,

j u ,v

i ,

j ,u ,v

= (i ,u )

(j ,v) u = cos i + sin jv = sin i + cos j

x(cos i + sin

j ) +y( sin i + cos j) = (x p)i + (y q)j

i

j

x cos y sin = x px sin + y cos = y q .

x

y

x = (x p)cos + (y q)sin y = (x p)sin + (y q)cos .

(x, y) x, yp q

(x, y) x, y, p q


30/106

x

y

=

cos sin

sin cos

x py q

.


31/106

a + ib

a, b R i

C = {a + ib|a, b R} .

C

a + ib (a, b)

a + ib

ai + b

j .

z=a + ib

a= Re z

z

b= Im z

z Re z Im z

{z C | Im z = 0}

z= a +bi

|z| = z z=

a2 + b2 .

C

R {z C | Im z= 0}


32/106

i i = 1

: (a + bi) + (c+ di) = (a + c) + (b+ d)i

: (a + bi) (c+ di) = (ac bd) + (ad + bc)i

a bi

z=a + bi

z

z+ z = 2a = 2Re z

z z = 2ib = 2iIm zz z = a2 + b2 =|z|2

z

z=

|z|2 z 1|z|2z= 1 z1 =

1

|z|2 z (a + bi)1 =

a

a2 + b2 b

a2 + b2i .

z

w=

zw

|w|2 a + bi

c+ di=

(a + bi)(c di)c2 + d2

z1+ z2 = z1+ z2

z1 z2 = z1 z2

(zn) = (z)n

|z| =|z| =| z||z1 z2| =|z1| |z2|

| |z1| |z2| | |z1+ z2| |z1| + |z2|

C

R

Ri

0


33/106

z= 0 z

0


34/106

z= 0 0


35/106

A, B, C, D

z1, z2, z1 z2 1

ODB OAC z2

z1

z3 z1 z3

z2

z1 z2 z1 z2

z1 z2 : z1= z2 : 1 .

z2

z1 Arg(z2)

|z2

|

z |z| = 1 Arg(z)

i

/2

i2 =1 /2

i2z= z .

z

= 0 z


36/106

z=r (cos +isin )

z1 =t(cos + i sin )

1 = z w

= r t (cos(+ ) + i( + ))

1

1

0

r t= 1 + = 0

z1 =1

r(cos() + i sin())

|z1| = 1|z| z1 arg z1 =

z1

Arg z1 =

2 Arg z Arg z= 00 Arg z= 0

z=r (cos + i sin )

z2 = r r (cos( + ) + i sin( + ))

= r2 (cos(2) + i sin(2))

z3

= r2

r (cos(2 + ) + i sin(2 + ))= r3 (cos(3) + i sin(3))

n

zn =rn (cos(n) + i sin(n)) .

De Moivre

z1

n N

zn =rn (cos(n) + i sin(n)) . De Moivre z= 0 n Z z= r(cos + i sin )

zn =rn (cos(n) + i sin(n))

z

S1

z1 = z .


37/106

n 1

n

n

n

1

1

n

n n n

zn = 1 .

z

z=r(cos + i sin )

zn =rn (cos(n) + i sin(n)) = 1 ,

rn = 1

cos(n) = 1

sin(n) = 0

n

cos + i sin

cos(n) = 1 sin(n) = 0 n= 2k k Z

= k

n2, k Z .

k =

2kn

k+n= k+ 2

cos k+n+ i sin k+n = cos k+ i sin k k= 0,1, . . . , n 1

0= 0, 1=2

n, . . . , k=

2k

n , . . . , n1=

2(n 1)n

n n

cos

2k

n + i sin

2k

n k= 0, 1, . . . , n 1

wn= cos 2n

+ i sin 2n

k= 0, . . . , n 1

wkn = cos2k

n + i sin

2k

n .

k= 0

1

n S1 = {z C | |z| = 1} 0

2n

n

n 1


38/106

a C

a= 0 n n a 0

zn = 0 z= 0 .

a = 0 a a= s(cos +isin ) s s1/n

n s z0 = s1/n

cos n

+ i sin n

zn0 =a z0 n a

zk n a zkz0

n= 1 zk

z0

n

a = 0 n n a= s(cos + i sin ) a n

zk =s1/n

cos

n+

2k

n

+ i sin

n+

2k

n

k= 0, 1, . . . , n 1

t+ i s+ i et(cos + i sin )

es(cos + i sin )

et(cos + i sin ) es(cos + i sin ) =et+s (cos( + ) + sin i( + ))


39/106

et(cos + i sin )

n

=ent (cos(n) + i sin(n)) ,

t + i et(cos + i sin )

et+i =et(cos + i sin ) ,

ez =eRe(z) (cos(Im z) + i sin(Im z)) .

z= elog |z|+iarg z .

ez ew =ez+w

(ez)n =enz

(ez)1 =ez

ez =ez

|ez| =eRe z arg(ez) = Im z

cos , sin

z= cos + i sin 1z

= cos isin

2cos = z+1

z

2i sin = z 1z

.


40/106

De Moivre zn = cos n+ i sin n zn = cos n isin n

2cos n = zn + 1zn

2i sin n = zn 1zn

cos6

26 cos6 =

z+

1

z

6

= z6 + 6z4 + 15z2 + 20 + 151

z2+ 6

1

z4+

1

z6

=

z6 +

1z6

+ 6

z4 +

1z4

+ 15

z2 +

1z2

+ 20

= 2 cos 6 + 12cos 4+ 30 cos2 + 20

cos6 = 1

32(cos6+ 6 cos 4 + 15 cos2 + 10)

(2i)5 sin5 =

z 1z

5

=

z5 1

z5

5

z3 1

z3

+ 10

z 1

z

25 sin5 = 2(sin5 5sin3 + 10 sin )

sin5

=

1

16 (sin5 5sin3 + 10sin ).

cos n, sin n

(cos6 + i sin 6) = (cos + i sin )6

= cos6 + 6i cos5 sin + 15i2 cos4 sin2

+20i3 cos3 sin3 + 15i4 cos2 sin4

+6i5 cos sin5 + i6 sin6 ,


41/106

cos6= cos6

15cos4 sin2 + 15cos2 sin4

sin6

sin6= 6 cos5 sin 20cos3 sin3 + 6 cos sin5 .

nk

k= 0, 1, . . . , n

(1 +x)n =

nk=0

nk

xk ,

nk

=

n!

k!(n k)! =(n k+ 1) (n k+ 2) (n 1)n

1 2 k .

0! = 1 n0

=nk

= 1

C =n

k=0

nk

cos k

= 1 +n cos +n(n 1)

2 cos2+ . . .+ cos n

S =n

k=0

nk

sin k

= n sin + n(n 1)2

sin2 + . . .+ sin n.

C S

(1 +ei)n

(1 +ei)n =n

k=0

nk

eik

=n

k=0 n

k(cos k+ i sin k)

= C+ iS.


42/106

1 + ein = (1 + cos + i sin )n=

2cos2

2+ 2i sin

2cos

2

n

=

2cos

2

ncos

2+ i sin

2

n

=

2cos

2

ncos

n

2 + i sin

n

2

.

C= 2cos2n

cosn

2

S=

2cos

2

nsin

n

2 .

n

Q= 1 + wn+ w2n+ . . .+ w

n1n .

wnQ = wn(1 + wn+ . . .+ wn1n )

= wn+ w2n+ . . .+ w

n1n + w

nn

= Q

(wn 1)Q= 0 wn= 1

1 +wn+ w2n+ . . .+ w

n1n = 0.


43/106

E2

E3

AB A E3

AB

AB A

B

ABB A


44/106

O, A, B, C

OA, OB, OC

A

A, B, C

AB, AC, AA

u = OA v = OB w = OC

z O z =OD a, b, c

R

z =au + bv + cw .

D z

u v u ,v ,w w OC C

c

OC = c

OC

A

B

OA

OB

a b OA = a OA OB = b

OB


45/106

OA, OB , OC

D

OD =

OA+

OB +

OC

= a u + b v + c w .

u ,v ,w

u ,v ,w

u =OA v =OB

u v prvu u v

prvu =OA A A

OB

u v = |v |(prvu ) (prvu ) OB v

(u ,v) u =OA v = OB AOB O, A, B

u =OA v =OB OA OB OADB


47/106

ON

(OA,

OB)

OA OB 0< <


48/106

(OB,

OA)

(OA,OB)

(OA,

OB)

(OA,

OB)

ON (

OA,

OB,

ON)

OX

(OA,

OB)

OA, OB

OA,OB

OA OB =

OO

OA,

OB

OX

OA,

OB

x ,y ,z O a

x y = y x

(a x ) y =a(x y)


49/106

x (y + z) = (x y) + (x z)

E3 O

O u =OA v =OB w =OC

u ,v ,w (,u ), (,v), (,w ) O (,u ),(,v),(,w )

(O, u ,v ,w )

z O

z =au + bv + cw .

(a, b, c)

z (O,u ,v ,w )

au , bv , cw

z

(O,u ,v ,w ) D OD D

OD

D

(O,

i ,j ,

k)

i ,

j ,

k

(i ,

j ,

k)

z z

z = (z i )i + (z j)j + (z k)k ,

z =z1i + z2j + z3k y =y1i + y2j +y3

k

z y =z1y1+ z2y2+ z3y3

|z | = z21+ z22+ z23.


50/106

(O,

i ,j ,

k)

i ,j

i j = k = j i

j k = i = k j

k i = j = i k .

z y = (z1i + z2j + z3k) (y1i + y2j + y3k)= z1y1

i i + z1y2i j + z1y3i k

+z2y1j i + z2y2j j + z2y3j k

+z3y1k i + z3y2k j + z3y3k k

= z1y1OO+ z1y2

k z1y3j

z2y1

k + z2y2

OO+ z2y3

i

+z3y1j z3y2i + z3y3OO ,

z y = (z2y3 z3y2)i + (z3y1 z1y3)j + (z1y2 z2y1)k .

33 a1, a2, a3 b1, b2, b3 c1, c2, c3

a1 a2 a3

b1 b2 b3

c1 c2 c3

=a1b2c3+ b1c2a3+ c1a2b3 a3b2c1 b3c2a1 c3a2b1 .

z y =

i

j

k

z1 z2 z3

y1 y2 y3

.


51/106

x ,

y ,

z

x

(

y

z)

x (y z) = (x1i + x2j + x3k)

(y2z3 y3z2)i + (y3z1 y1z3)j + (y1z2 y2z1)k

= x1y2z3 x1y3z2+ x2y3z1 x2y1z3+ x3y1z2 x3y2z1

=

x1 x2 x3

y1 y2 y3

z1 z2 z3

.

x ,y ,z

x ,y ,z

(x ,y ,z) (x ,y ,z)

(x y) z = x (y z) .

(x y) z = z (x y)

=

z1 z2 z3

x1 x2 x3

y1 y2 y3

=

x1 x2 x3

y1 y2 y3

z1 z2 z3

= x (y z) .

[xyz] = x (y z) = (x y) z .

[xyz] = x (y z)=

x

(z

y)

= [zyx ] .


52/106

b = x (y +z)x yx z b 2 = 0

b = 0

b b = b (x (y + z) x y x z)

=

b x (y + z) b x y b x z= (

b x ) (y + z) (b x) y (b x ) z

= (b x ) ((y + z) y z)

= 0 .

[xyz] = [yzx ] = [zx y] ,

[zyx ] = [yxz] = [xzy] .

x (y z) y z

y ,z x (y z)

y

z

x

(y

z) =

i

j

k

x1 x2 x3

y2z3 y3z2 y3z1 y1z3 y1z2 y2z1

=

i (x2(y1z2 y2z1) x3(y3z1 y1z3))+

j (x3(y2z3 y3z2) x1(y1z2 y2z1))

+k (x1(y3z1 y1z3) x2(y2z3 y3z2))

=

i (y1(x1z1+ x2z2+ x3z3) z1(x1y1+ x2y2+ x3y3))+

j (y2(x1z1+ x2z2+ x3z3) z2(x1y1+ x2y2+ x3y3))

+k (y3(x1z1+ x2z2+ x3z3)

z3(x1y1+ x2y2+ x3y3))

= (x z)y (x y)z .


53/106

x (y z) = (x z)y (x y)z

(x y) z = (x z)y (y z)x .

u [u ]

u

[u ] = {v v u } .

OC=OA OB [

OA] [

OB]

[OA] [OB] = [OA OB]

= [OC] .

u OA,v OB w OC

[

u ]

[

v] = [

w ] .

(O,

i ,j ,

k)

(P,u ,v ,w )

(O,

i ,j ,

k) (O,u ,v ,w )


54/106

X (x, y, z)

(O,

i ,j ,

k) X

OX = x

i + y

j + z

k .

(x, y, z) X

(O,u ,v ,w ) OX = xu + yv + zw .

i ,

j ,

k

u ,v ,w i = a1

u + b1v + c1wj = a2

u + b2v + c2w

k = a3u + b3v + c3w

OX = x(a1

u + b1v + c1w )+y(a2

u + b2v + c2w )+z(a3

u + b3v + c3w )= (xa1+ ya2+ za3)

u+(xb1+ yb2+ zb3)

v+(xc1+ yc2+ zc3)w


55/106

x = a1x+ a2y+ a3z

y = b1x + b2y+ b3z

z = c1x + c2y+ c3z

x

y

z

=

a1 a2 a3b1 b2 b3

c1 c2 c3

xy

z

(O,

i ,

j ,

k)

(P,u ,v ,w ) OP =x0

i + y0

j + z0

k ,

P

(O,i ,j ,k)

(x0, y0, z0)

P X =

OX OP

= (x x0)i + (y y0)j + (z z0)k .

(x, y, z) X (P,u ,v ,w )

P X=xu + yv + zw ,

x = a1(x x0) +a2(y y0) +a3(z z0)y = b1(x x0) + b2(y y0) +b3(z z0) z = c1(x x0) + c2(y y0) +c3(z z0)

i ,

j ,

k

u ,v ,w

u ,v ,w i ,j ,k


56/106

u u (O,

i ,

j ,

k) u

i ,

j ,

k

u = (

u

i )i + (

u

j)j + (

u

k)

k .

i u u i

1 u i

u j u k 2, 3 u j ,k

u i = cos 1 u j = cos 2 u k = cos 3

a1= i u = u i = cos 1

a2 = cos 2 a3 = cos 3

u =a1i + a2j + a3k .

(a1, a2, a3)

u (O,i ,j ,k)

v =b1i + b2j + b3k


57/106

w =c1i + c2j + c3k (O,i ,j ,k) (P,u ,v ,w )

(O,

i ,

j ,

k)

u = a1i + a2j + a3kv = b1i + b2j + b3kw = c1i + c2j + c3k ,

i

j

k

u a1 a2 a3v b1 b2 b3w c1 c2 c3

(O,

i ,

j ,

k)

(P,u ,v ,w ) P (O,i ,j ,k) (x0, y0, z0)

X (x, y, z) (O,

i ,j ,

k)

(x, y, z)

X

(P,u ,v ,w )

x = a1(x x0) +a2(y y0) +a3(z z0)y = b1(x x0) + b2(y y0) +b3(z z0)z = c1(x x0) + c2(y y0) +c3(z z0)


58/106

(x, y) f : R2

R

P(x, y)

f(x, y) = 0 {(x, y) R2 : f(x, y) = 0} f(x, y) = 0

{(x, y) R2 : f(x, y) = 0}

f(x, y) = 0

f(x, y) = ax+by + c

ax+by + c = 0


59/106

{(x, y) : ax + by + c= 0}

=a/b y

c/b

f(x, y) =x2 +y2 r2 r >0

f(x, y) = 0

x2+y2 =r2 r (0, 0)

O r

C

(a, b) P (x, y)

P

C

CP

= 3

(x a)2 + (y b)2 = 3

f(x, y) =

(x a)2 + (y b)23 (x a)2 + (y b)2 x y (x a)2 + (y b)2 = 9 g(x, y) = (x a)2 +(y b)2 9

(x a)2 + (y b)2 = 0

P

(a, b)

f(x, y) g(x, y) R2

f(x, y)g(x, y) f

g f(x, y)g(x, y) = 0

f(x, y) = 0 g(x, y) = 0

y2 x2 = 0

(y+x)(y x) = 0

f(x, y) g(x, y) R2

f(x, y)2 +g(x, y)2

f g f(x, y)2 +

g(x, y)2 = 0 f(x, y) = 0

g(x, y) = 0

f(x, y) = 0

g(x, y) = 0


60/106

P (x, y, z)

f : R3 R P(x, y, z)

f(x, y, z) = 0

C(a, b, c) r

(x a)2 + (y b)2 + (z c)2 =r2

g(x, y, z) = ax+by +c

f(x, y)

ax+ by +c = 0

P(x, y, z)

z R z

f(x, y, z)

z (x, y) ax+

by+ c= 0

x2

+ y2

+ z2

= rax+ by+ cz = d

y2 +z2 = 4

{(x, y, z) R3 : y2 + z2 = 4}

x (y, z)

O

2

{(x, y) R2 : 2x 3y+ 2 = 0}

{(x, y) R2 : y= 1}

{(x, y) R2 : (y 1)(2x y) = 0}

{(x, y) R2 : x2 + y2 2x 2y 2 = 0}

{(x, y) R2 : x2 + y2 = 2xy}


61/106

{(x, y, z) R3 : x2 + y2 + z2 + 2x 2y 2 = 0}

{(x, y, z) R3 : x2 + y2 + z2 + 2x 2y+ 2 = 0}

{(x, y, z) R3 : 2x 3y= 0}

{(x, y, z) R3 : y= 1}

{(x, y, z) R3 : (y 1)(x2 + z2 1) = 0}

f : R R2 f(t) = (x(t), y(t)) t R P(x(t), y(t))

f : R R3 f(t) = (x(t), y(t), z(t)) t (x(t), y(t), z(t))

f : R2 R3 f(s, t) = (x(s, t), y(s, t), z(s, t))

(s, t) R2

P(x(s, t), y(s, t), z(s, t))

A(1,2)

B(1, 3) f(t) = (1 + 2t, 3 t)

(a, b)

r

0


62/106

s t s2 + t2 1

x(t) = 1 t y(t) = 2 3t

t

t= 1x

t= y23

t

1 x= 13

(y 2) 3x+ y 5 = 0

|z w1| = |z w2|, > 0 .

w1 w2

z

|z w1| =|z w2|, >0 .

= 1

|z

w1

|=

|z

w2

|

z

w1 w2

= 1 z w1 w2

p w1 w2

p

w1 w2

(w1, w2, p) = p w1w2 p,

(w1, w2, p) = = (w1, w2,p) . p p

p=w1+ w2

1 + p =

w1 w21

z

|z

w1

|=

|z

w2

|

z w1, w2


63/106

|z w1| = 2|z w2|

p p

c = p+p

2

= (w1+ w2)(1 ) + (w1 w2)(1 +)

2(1 2

)

= w1 2w2

1 2

r = |p p|

2

= |(w1 w2)(1 ) (w1 w2)(1 +)|

2|1 2

|=

|| |w1 w2||1 2|

c r

(z c)(z c) =r2

z(1 2) w1+ 2w2

z(1 2) w1+ 2w2

= 2(w1 w2)( w1 w2)


64/106

zz

zw1

zw1+ w1w1 =

2(zz

zw2

zw2+ w2w2)

|z w1|2 =2|z w2|2 .

= 1


65/106

P(x1, y1, z1) Q(x2, y2, z2) R(x3, y3, z3) .

X(x, y, z)

P X

P Q,

P R s t

P X=s

P Q + t

P R

(x x1, y y1, z z1) =s(x2 x1, y2 y1, z2 z1) +t(x3 x1, y3 y1, z3 z1)

(x, y, z) = (1

s

t)(x1, y1, z1) +s(x2, y2, z2) +t(x3, y3, z3) .

P Q

P R

P Q P R

P Q P R P X= 0 .

x2 x1 y2 y1 z2 z1x3

x1 y2

y1 z3

z1

x x1 y y1 z z1 = 0


66/106

A = (y2

y1)(z3

z1)

(z2

z1)(y3

y1)

B = (z2 z1)(x3 x1) (x2 x1)(z3 z1)C = (x2 x1)(y3 y1) (y2 y1)(x3 x1)

Ax+ By + Cz (Ax1+ By1+ Cz1) = 0

P(x1, y1, z1)

n (k, , m) X(x, y, z) P X n

P X n = 0 ,

(x x1, y y1, z z1) (k, , m) = 0 ,

kx+ y+ mz (kx1+ y1+ mz1) = 0 .

Ax+ By + Cz+ D= 0 .

(x1, y1, z1)

Ax1+ By1+ Cz1+ D= 0 ,

(x, y, z)

A(x x1) +B(y y1) +C(z z1) = 0 ,

(x x1, y y1, z z1) (A, B, C) (x, y, z) (x1, y1, z1)

(A, B, C)

B=C= 0 A = 0

x= DA

O y z

A= 0, BC= 0

By + Cz+ D= 0


67/106

Ox

O y z By + Cz+ D= 0

D= 0

Ax+ By + Cz= 0

O

ABCD= 0 (DA

, 0, 0)

(0,DB

, 0) (0, 0, DC

)

x, y, z , ,

= 0 x

+y

+z

= 1 .

P(x1, x2, x3) a = (u, v, w)

(x, y, z) = (x1, y1, z1) +s(u, v, w)

s uvw = 0x x1

u =

y y1v

=z z1

w

vx uy (vx1 uy1) = 0wy vz (wy1 vz1) = 0 .

w= 0 uv= 0 s

vx uy (vx1 uy1) = 0z = z1 .

A1x + B1y+ C1z+ D1= 0 ,

A2x + B2y+ C2z+ D2= 0 .

A1A2= B1B2

= C1C2= D1D2


68/106

A1A2

=B1B2

=C1C2

= D1D2

(A1, B1, C1) =

k(A2, B2, C2)

A1A2

= B1B2

A1A2

= C1C2

,

(A1, B1, C1) (A2, B2, C2)

u = (A1, B1, C1) (A2, B2, C2)

(x, y, z) = (x0, y0, z0) +t(A1, B1, C1) (A2, B2, C2)

(x0, y0, z0)

A1x0+ B1y0+ C1z0 =

D1

A2x0+ B2y0+ C2z0 = D2 .

A1B2 B1A2= 0 z0= 0

A1x0+ B1y0 = D1A2x0+ B2y0 = D2

x0=(D1B2 B1D2)

A1B2 B1A2, y0=

(A1D2 D1A2)A1B2 B1A2

.

Ax+ By +C z+D = 0 n = (A, B, C) X0 X1

X1

X0X1

n

d = (prn X0X1) = X

0X

1 n

|n |


69/106

= (x1 x0)A+ (y1 y0)B+ (z1 z0)C

A2 + B2 + C2

=

Ax1+ By1+ Cz1

(Ax0+ By0+ Cz0)

A2 + B2 + C2

Ax0+ By0+ Cz0 = D

d=Ax1+ By1+ Cz1+ D

A2 + B2 + C2

d X1

n = (A, B, C)

OX =

OX0 + ta (x, y, z) =

(x0, y0, z0) +t(a, b, c) X1 OX1= (x1, y1, z1)

X0X1 a

X1

e = (X0X1 a) a

X1 X0X1

e

d = |preX0X1| =

e X0X1

|e

|

= |(X0X1 a) a X0X1|

|(X0X1 a) a |.

a

X0X1 a

d = |(X0X1 a) (a X0X1)|

|X0X1 a | |a |

= |

X0X1

a

||a | .


70/106

1, 2

OX =

OX1+ t

a1 t ROX =

OX2+ s

a2 s R

X1X2,

a1 a2

[X1X2,

a1 ,a2 ] = 0 .

u a1 a2

u = a1 a1.

X1X2

u

d = |pra1a2X1X2|

= |X1X2 a1 a2 ||a1 a2 |

= |[X1X2,a1 ,a2 ]|

|a1 a2 | .

T1 T2 1

2 T1 1

OT1 =

OX1+ t

a1

T2 2

OT2 =

OX2+ s

a2.

T1T2

a1 a2 T1T2 =

a1 a2.

T1T2=

OT2 OT1

OX2 OX1+ s a2 t a1 = a1 a2,


71/106

(s, t, ) t, s

T1, T2

1 2 1

A(2,1,0)

u = (1, 3,1)

2

B(0,1,3) v = (2, 1, 1)

[AB,u ,v] =

2 2 31 3 12 1 1

= 29 .

u v = (1, 3,1) (2, 1,1) = (4,3,5) ,

|u v | = 52

d=|[AB, u ,v]|

|u

v|

=| 29|

5

2=

29

2

10 .

OB OA+ s v t u = (u v) ,

(2, 2, 3) +s (2, 1, 1) t (1, 3,1) = (4,3,5) ,

2s t 4 = 2s 3t + 3 = 2s + t + 5 = 3 ,

s t

t= 1

25 , s= 7

50.

T1 = (2,1, 0) + 125

(1, 3,1)

T2 = (0,1,3) 750

(2, 1, 1)


72/106

|T1 T2| = 1

50 |116,87,145| =29

2

10 .


73/106

r >0 C

X

|CX| =r

C

x2 + y2 =r2

C : (x0, y0)

(O,

i ,j )

(x x0)2 + (y y0)2 =r2

(x, y) = (x0, y0) +r(cos , sin ) 0


74/106

(x, y) = (x0, y

0) +r1

t2

1 +t2,

2t

1 +t2

t (,) (x0 r, y0) t

1 t21 +t2

, 2t

1 +t2

t

t= sin

1 + cos .

r X1: (x1, y1)

X : (x, y)

(x, y) = (x1, y1) + t(a, b)

(x1+ ta)2 + (y1+ tb)

2 =r2 ,

x21+ y21+ 2t(ax1+ by1) + t

2(a2 + b2) r2 = 0 . t

= (ax1+ by1)2 (a2 + b2)(x21+ y21 r2) .

< 0

> 0

= 0

X1 x21+ y

21

r2 = 0 X1

ax1+ by1 = 0 ,


75/106

X1

(a, b) = (y1, x1)

(x, y) = (x1 ty1, y1+ tx1) X1 : (x1, y1)

O r

x1x + y1y= r2

X1 X1

x21 +y21 r2 0 a, b X0 : (x0, y0)

X1

X0 x0x + y0y=r

2

X1 X0 x0x1+ y0y1 = r2

X1

xx1+ yy1 = r2 .

x21+ y21 r2 >0

X0 X0

X0, X0

X1

x1x+ y1y= r2 ,

X1

X1

X1 X1

X1

X1

X1

(x, y)

X1: (x1, y1)

X1


76/106

Ax+ By + C= 0

(x1, y1) =

Ar

2

C ,Br

2

C

.

C : (x0, y0)

F1, F2

F1F2 i =

F1F2

|F1F2| F1, F2

(C, 0) (C, 0) (O,i ,j)

X

F1, F2

2a

a > c

|F1X| + |F2X| = 2a ,

(x+ c)2 + y2 +

(x c)2 + y2 = 2a

(x c)2 + y2 =

a ca

x

a2 c2a2

x2 + y2 =a2 c2

a > c a2 c2 >0 b2 =a2 c2 x2

a2+

y2

b2 = 1 .

c= 0 a2 =b2

a= c

F1F2

e= c

a

1 : x= a2

c 2 : x=

a2

c


77/106

X r1, r2 X

F1, F2

r21 = (x + c)2 + y2 , r22 = (x c)2 + y2 .

r21 r22 = 4cx r1+r2 = 2a r1 r2 = 2cx

a r1 =a +

cx

a

r2=a cxa

X

d1=d(X, F1) =a2

c + x d2 = d(X, F2) =

a2

c x ,

r1d1

=a2 + cx

a

c

a2 + cx=

c

a=e

r2d2

X1 : (x1, y1) X2 : (x2, y2) X : (x, y)

X1X2 X

x=

x1+ tx21 +t , y=

y1+ ty21 +t ,

t= (X1X2X)

(X1X)

(XX2)

(x, y) t

(x1+ tx2)2

a2(1 +t)2 +

(y1+ ty2)2

b2(1 +t)2 = 1 ,

t = 1

x21+ 2tx1x2+ t2x21a2

+y21+ 2ty1y2+ t2y22

b2 = 1 + 2t+ t2 ,

x22a2

+y22b2

1

t2 + 2x1x2

a2 +

y1y2b2

1

t +

x21a2

+y21b2

1

= 0

L= x2

2

a2+

y22

b2 1 M= x1x2

a2 + y1y2

b2 1 N= x21

a2+

y21

b2 1

M2 LN= 0 .


78/106

X1X2

X1

M= 0

X1

x1x

a2 +

y1y

b2 = 1 .

X1 x21a2

+y2

b >1

x1x

a2 +

y1y

b2 1

2

x21a2

+y21b2

1x2

a2+

y2

b2 1= 0

X1

X1

F1X1F2

n

Q1 cos(X1F1,

n )

cos(X1F2,

n )

F1, F2 x1x

a2 +

y1y

b2 = 1 (x1, y1) b

x2

a2 + y2

b2 = 1

(x, y) = (a cos , b sin ) .

(a,0)

(x, y) =

a(1 t2)

1 + t2 ,

2bt

1 +t2

.

t


79/106

F1, F2

F1F2 i =

F1F2

|F1F2| F1, F2

(c, 0) (c, 0) X |F1X| |F2X| = 2a c > a

(x+ c)2 + y2

(x c)2 + y2 = 2a

(x c)2 + y2 = cxa a

a2 c2a2

x2 + y2 =a2 c2 .

c > a b2 =c2

a2

x2

a2 y

2

b2 = 1 .

x2

a2 1 = y

2

b2 >0

|x| > a

y2

b2 x

2

a2 = 1

(0, c) (0,c)

e= c

a

1 : x= a2

c 2 : x=

a2

c


80/106

X : (x, y) =

x1+ tx2

1 +t ,

y1+ ty21 +t

X1 : (x1, y1) X2 : (x2, y2)

t

(x1+ tx2)2

a2(1 +t)2 (y1+ ty2)

2

b2(1 +t2) = 1 ,

x22

a2

y22

b2

1 t2 + 2

x1x2

a2

y1y2

b2

1 t +

x21

a2

y21

b2

1= 0 .

L= x2

2

a2 y22

b2 1 M= x1x2

a2 y1y2

b2 1 N= x21

a2 y21

b2 1

M2 LN= 0 X1X2

X1

M= 0

X1 x1x

a2 y1y

b2 = 1 .

X1 x21a2

y21

b2


81/106

x2

a2y2

b2 = 1

cosh2 t sinh2 t = 1

(x, y) = (a cosh s, b sinh s) , s R x >0

(x, y) = (a cosh s, b sinh s) , s R

x


82/106

(y22 2px2)t2 + 2(y1y2 p(x1+ x2))t + y21 2px1= 0 .

L= y22 2px2 , M=y1y2 p(x1+ x2) , N=y21 2px1 .

M2 LN= 0

X1X2

X1

M= 0

X1

y1y= p(x1+ x)2px1

(y1y p(x1+ x))2 (y21 2px1)(y2 2px) = 0

X1

X0

X0F X0

u X0 cos(

n ,

X0F) cos(

n ,

i )

x= 2pt2

(x, y) = 2p(t2, t) .


83/106

f(x, y) =Ax2 + 2Bxy + Cy2 + 2Dx+ 2Ey + F= 0

A, B, C

(x, y) (0, 0)

(x,y)

Ax2 + 2Bxy + Cy2 + 2Dx+ 2Ey + F =

= A(x)2 + 2B(x)(y) +C(y)2 + 2D(x) + 2E(y) + F .

4Dx+ 4Ey = 0 .

(x1, y1), (x2, y2), (x3, y3)

D E

D= E= 0 .

y

(x, y) (x, y)

Ax2 + 2Bxy + Cy2 + 2Dx+ 2Ey + F =

= A(x)2 + 2B(x)y+ Cy2 + 2D(x) + 2Ey + F

4Bxy + 4Dx= 0 ,


84/106

B=D = 0 .

P(x0, y0)

(x, y)

(x, y)

x = x0 + x cos y sin y = y0 + x

sin + y cos .

f(x, y)

f(x, y) =Ax2 + 2Bxy+ Cy2 + 2Dx+ 2Ey+ F = 0 ,

A, B, . . . , F A, B , . . . , F, x0, y0,

J1 = A+ C=A+ C

J2 = AC B2 =AC B2

J3=

A B D

B C F

D E F

=

A B D

B C E

D E F

.

J1, J2, J3

J2= 0 J2= 0 A C

A+ C =A+ C

J2= 0 J2= 0


85/106

D = (Ax0+ By0+ D)cos + (Bx0+ Cy0+ E)sin

E = (Ax0+ By0+ D)sin + (Bx0+ Cy0+ E)cos

x0, y0 D

= E = 0

(x0, y0) sin cos

D =E = 0

Ax0+ By0+ D = 0

Bx0+ Cy0+ E = 0

J2 = AC

B2

= 0

x0= DC BEAC B2 y0 =

AE DBAC B2 .

(x0, y0) D=E = 0

Ax2 + 2Bxy+ Cy2 + F = 0

B = 0

B

B = A sin cos + B(cos2 sin2 ) +Csin cos

B = 0

B(cos2

sin2

) = (A C)sin cos

2B cos2= (A C)sin2

A= C

B= 0

cos2= 0

=

4

A =C

tan2=

2B

A C


86/106

4

< 4

2

0 x2

a2+

y2

b2 = 1

AF < 0, CF > 0 x2

a2 y

2

b2 = 1 x

AF > 0, CF < 0 x2

a2 y

2

b2 = 1 y

F = 0

Ax2 + Cy2 = 0


87/106

y = A

C

x

AC0 y= ix

F

A, F

C J1, J2, J3

FA FC = F2

AC = F2

AC B2 = F2

J2.

J3 =ACF =J2F

J3= 0 J2 >0 J1J3 < 0 J1J3 > 0

J3= 0 J2 0

J3 = 0 J2


88/106

A C= 0

=

4

cos = sin =22

x= 1 +2

2 x

22

y =2

2 (x y

2)

y = 2 +

2

2 x+

2

2 y =

2

2 (x+ y+ 2

2)

1

2(x y

2)2 4

2

2 (x y

2)

2

2 (x+ y+ 2

2)+

+1

2(x+ y+ 2

2)2 + 10

2

2 (x y

2) 8

2

2 (x+ y+ 2

2) + 7 = 0 ,

x2 + 3y2 6 = 0 ,

y

x2

6 y2

2 = 1 .

J1=A+ C, J2 = AC

A, C

t2 J1t + J2

t2 2t 3 = 0

(t + 1)(t 3) = 0

A = 1, C = 3 A = 3, C= 1


89/106

J2= 0

J2 = 0

xy B = 0

J2 =AC = 0 A, C

C= 0

Ax2 + 2Dx+ 2Ey+ F = 0 .

E

= 0

x+

D

A

2= 2E

A

y+

F

2E D

2

2AE

,

D

A, D

2AF2AE

x =DA

E

A

x =x+D

A, y = y D

2 AF2AE

.

p= EA

x, y x, y

x2 = 2py

B = C = 0 J3 = AE2 J1 = A

p2 =E2

A2 = J3

J13.

E = 0

x=D D2 AF

A

D2 AF > 0 D2 AF = 0 D2 AF


90/106

D= 0

y+

E

C2

= 2D

C

x+

F

2D E2

2CD

E2CF2CD

,EC

y = EC

p= DC

y2 = 2px .

p2 = D2

C2 = J3

J13.

D = 0

y=E E2 CF

C

E2 CF > 0

E2 CF = 0

E2 CF0

J3= 0 J3= 0 J3= 0


91/106

O OA

X r

O

= (OA, OX)

(r, ) X

(r, + 2k)

k

Z

x = r cos

y = r cos

r = x2 + y2

= arctany

x,

x y

(, ] (

3,1)

r=

3 + 1 = 2 , = arctan 1

3=

6.

(

3, 1),(

3,

1) (

3,

1)

2, 5

6

,

2,56

2,

6


92/106

O a

r= a .

u

O

t

r= ut

= t

t

r= u

.

O

x2 + y2 + z2 =2

X : (x, y, z) X X

Oxy

X : (x, y, 0)

OX

x OX

= (Ox, OX)

= (OX, OX) , 2

2

X

X

(, , )

X

=

x2 + y2 + z2


93/106

= arctany

x, <

= arcsin zx2

+ y2

+ z2

,

2

2

x = |OX| cos = cos cos y = |OX| sin = cos sin z = (XX) = sin

O a

= a

= 0

0 = 0 (x, y) 0= 0

z

2 0 0 >0 2 +0 0


94/106

z

cos =

x2

+ y2

.

X

(x, y, z)

X X (x, y) (x, y, 0) (x, y)

X

(, )

r=

x2 + y2 , = (Ox, OX)

(r, , z) X

r =

x2 + y2

= arctany

x, <

z = z

x = |OX| cos = r cos y =

|OX

|sin = r sin

z = z

z a

r= a .

z=c


95/106

(x, y) |c|

= 0

z

= a , = 0

z

= az , a = 0

z z

z


96/106

z

(x, y, z)

x2 + y2 =a2 .

z

R

z

z=x y = 0

x2 + y2 =z2

2.


97/106

z

(x, z)

f(x, z) = 0 , y= 0 , x 0 , (x, y, z) R3

f(

x2 + y2, z) = 0.

(x0, 0, z0)

z = z0

x0 z

(x, y, z0) x2 + y2 =x0

x2

a2+

y2

b2 = 1, z= 0

x

x2

a2+

y2 + z2

b2 = 1

z

(x a)2 + z2 =r2 , y = 0 , a > r >0 .


98/106

x2 + y2 a2

+ z2 =r2

x2 + y2 2pz= 0

f(

x2 + y2, z) = 0

x2 = 2pz

y = 0

z

x

y

z

x2

a2+

y2

b2+

z2

c2 = 1 , a, b, c >0 .

f(x, y, z) =x2

a2+

y2

b2 +

z2

c 1

f(x, y, z) =f(x, y, z) =f(x,y, z) =f(x, y,z)


99/106

(y, z) (x, z)

(x, y)

2a,2b,2c

A : (a,0, 0)

A : (a,0, 0)

B : (0, b, 0)

B : (0,b, 0) C : (0, 0, c) C(0, 0,c) (x, y)

x2

a + y

2

b = 1 , z= 0 .


100/106

(x, y) z= k c k c

x2

a2+

y2

b2 = 1 k2

c2, z=k

|k| =c |k| < c

a

1 k

2

c2 , b

1 k

2

c2 .

(x, z) (y, z)

a, b, c

x2a2

+y2b2

z2c2

= 1 , a, b, c >0 .

(x, y) (x, z)

(y, z) x, y, z

(x, y)

x2

a2+

y2

b2 = 1 , z= 0 .

(x, z)

x2

a2+

z2

c2 = 1 , y= 0 ,

(y, z)

y2

b2+

z2

c2 = 1 , x= 0 .

z=k

x2

a2+y

2

b2 = 1 k

2

c2 , z=k .


101/106

x= m |m| =a

y2

b2z2

c2 = 1 m2

a2 , x= m .

|m| =a 1 m2a2

= 0

y

b = z

c, x= m ,

(m,0,0)

a= b

x2

a2 y

2

b2 +

z2

c2 = 1 , a, b, c >0 .

(x, y) (x, z) (y, z)

x, y, z


102/106

(x, y)

z=k

|k| > c |k| =c

(x, z) (y, z)

a= b

x2

a2

+y2

b2

= 2cz , a, b >0 , c

= 0 .

(x, z)

(y, z)

z

(x, y)

(x, y) z = k k > 0

x2

a2+

y2

b2 = 2ck , z =k .

k


103/106

(x, z) (y, z)

(x, z)

x2 = 2a2cz , y = 0 ,

y=m

x2

a2 = 2cz m

2

b2 , y= m .

x2

a2 y

2

b2 = 2cz , a, b >0 , c = 0 .

(x, z) (y, z)

z

(x, y)

x2

a2 y

2

b2 = 0 , z= 0 ,


104/106

(x, y) z=k

x2

a2 y

2

b2 = 2ck , z = k .

(x, z) (y, z)

x= m

y2

b2 = 2cz+m2

a2 , x= m .


105/106

x2

a2+

y2

b2 z

2

c2 = 0

y2 = 2cz, c = 0

y2

b2 z

2

c2 = 1

y2

b2 +

z2

c2 = 1

x2

a2y2

b2 = 0

x2

a2+

y2

b2 z

2

c2 = 0

z2 = 0

x2

a2+

y2

b2 = 0

x2a2 +

y2

b2 + z2c2 = 0

x2

a2+

y2

b2+

z2

c2 = 1

x2 y2 z2 + 4x 8y+ 2z 17 = 0

(x + 2)2 4 (y+ 4)2 + 16 (z 1)2 + 1 17 = 0

(x+ 2)2 (y+ 4)2 (z 1)2 = 4 . (2,4, 1)

x2

4 y

2

4 z

2

4 = 1 .

(

2,

4, 1) x

(0,4,1) (4,4,1)


106/106

(x, y), (x, z), (y, z)

x = 4 y = 4

z= 4

i 9x2 + 4z2 = 36y

ii 4y2 + 4z2

x2 = 0

iii 9x2 y2 = 4z

iv 2y2 + 4z2 =x2

v x2 + y

2

4 + z

2

9 = 1

vi x2

9 +

y2

16z

2

4 = 1

vii

x2

4 y2

9 + z2 = 1

viii

x2

4 y

2

9 = 0

x2 + y2 + z2 6x+ 4y 8 = 0

Documents

Agma Notes