Высшая математика. Том 2

1.30. 3 &

%. . % + . . %

1: F& '-" " " 9 &@ @ ! " &

a11·x2 + 2a12·xy + a22·y2 + 2a13·x + 2a23·y + a33 = 0

= "> # ! & "$ & # C A& ! " ", & " #& #-

+ # '+" 2a′ xy .

12

B& ! & 3 < '+" +! = " ctg2ϕ = a11 − a22 .

2a12

2: F& '-" " " 9 &@ @ ! " &

a11·x2 + 2a12·xy + a22·y2 + 2a13·x + 2a23·y + a33 = 0

= "> # ! ' @ ! + " # C A& ! " ", & " #&

# + #& '+" 2a′ x 2a′ y .

13 23

3 +& & ! ' @ ! +& ! +"> Oy 3 < '+" - "# ! @ a11·x2 + 2a13·x a22·y2 + 2a23·y ! @

! # @ D ! & 9 + + #																																			( A A & 2a12·xy)
" " 9 &@ @ ! " &:																																											2
a x	2	+	2a x = a			(x		2	+ 2		a			x		+			a		2		− (		a			2			= a				(x +	a		2	−	a
											13						(		13	)					13		)			)						13	)			13
11				13	11						a11								a11						a11									11		a11				a11
a y			2	+ 2a y = a					( y	2	+				a			y + (				a			2	− (			a		23		2		= a	( y +			a			2	−		a	2	.
							22					2				23							23	)								)		)					23		)					23
	22				23										a22							a22							a22						22				a22						a22
										a13
					x′	= x +
										a11
C												.
										a23
					y′	= y +
										a22
																						)
3											9						&@ @													! " &						9									" "#:

3x2 + 4xy + 3y2 + 8x + 12 y + 1 = 0 ! + $@ 3 @ @ " &. N >&<# " " 9 &@ @ ! " &:

			I = 3 + 3 = 6 , I				2	=	3	2	= 9 − 4 = 5 ,
			1						2	3
I3 =		3	2	4		= 9 + 48 + 48 − 48 − 2 − 72 = −17 .
		2	3	6
		4	6	1

? A. 1.7. 3 <# 9 I2 > 0 , I1I3 < 0 – !+.

A "<# +" & # = # A& # # > – 4xy. " E'- @ A3 + D<# & ! &:

41

ctg2ϕ = 3 − 3 = 0 , 2ϕ = 90° , ϕ = 45° .

	4
	C, 6 #& " # @ " #& '							# 3 #:
		2				2
	x = x′cos 45° − y′sin 45° = x′				− y′			,
		2				2
			2				2
	y = x′sin 45° + y′cos 45° = x′				+ y′				.
		2				2

+ "<# " " # + ' x , ! # ! A :

3 (x′ − y′)2 + 4 (x′ − y′)(x′ + y′) + 3 (x′ + y′)2 + 4 2(x′ − y′) + 6 2(x′ + y′) + 1 = 0 .

2

2

2

3

x′2 − 3x′y′ +

3

y′2 + 2x′2 − 2 y′2 +

3

x′2 + 3x′y′ +

3

y′2 +

2

2

2

2

′

− 4 2 y

′

′

′

+ 1 = 0

4 2x

+ 6 2x + 6 2 y

′2

+ y

′2

′

+ 2 2 y

′

+ 1 = 0 .

5x

+ 10 2x

! A> ! A +" $> , " E' @ "<# !-$ :

′2

′

′2

′

+ 2 − 2) =

′

+

2)

2

− 10 .

5x

+ 10 2x =

5(x

+ 2 2x

5(x

y′2 + 2 2 y′ = y′2 + 2 2 y′ + 2 − 2 = ( y′ + 2)2 − 2 .

C # <#

′

2)

2

− 10

+ ( y

′

+

2)

2

− 2 + 1

= 0.

5(x +

E' @ " " C&<# 3 +& & " ! ' @ ! + "

x′′ = x′ +	2	′′2	+ y	′′2	= 11
	5x
y′′ = y′ +	2

A ! "3 6 # 3 @ " ", + 3 # <#:

	x′′2					y′′2
					+		= 1. O
(		5	)	2		( 11)2
	11

1.31. 4

3 " > &@ @ ! " & ! 3 # . " ' 9 3 M(x,y) &-

< '+" + ' OM=R. + &D3 6 #& & " A3 + " + # C # 3 #, -

C # " " (x − x0 )2 + (y − y0 )2 = R , A

(x − x0 )2 + (y − y0 )2 = R2 , " < '+" -

. ? #, " " E # ! 3 & # < @ "

x2 + y2 = R2 .

+. 1.28 B

42

)

? ! + " " , % ! > ' 3 3 & !(2,3) % # < E 3E (–6,8).

	N + &<# # ' " " (x − x0 )2 + (y − y0 )2 = R2 ,
(x0 ; y0 ) – E & .
	" A3 + " &+& $# C &
							OA . C
+ $@ :
			(xA − x0 )2 + (yA − y0 )2 =	(2 + 6)2	+ (3 − 8)2	= 82 + 52	= 89 .
	OA	=

! ': (x + 6)2 + (y − 8)2 = 89 . O

< '+" @ # 3 # +E 3 ! %, " " > + + $ > 6 +$ 3 F1 F2 E <9 ! % (, % ) < 3 ! + $ .

" " x2 + y2 = 1 - a2 b2

' 1 & + -

.

3 $ b - D ' ! D $ # D ! +"# !+.

3 F1(–c;0) F2( ;0) –

D '+" 6 &+ # !+,

	= "ε =	c	$@	+E -
		a
	+ #.		"	!+:
	c2 = a2 − b2 .
	? +. 1.29 a>c,			#& "
	!+ ε < 1.

+. 1.29 !+

)

+ 3 " " !+, " %: a = 9,

ε =

2

x2

y2

3

c

N

+

= 1 3 " "# !+. ? &

ε =

! <, %

a2

b2

a

2

c = aε . + "D3 ! 3 > # c = aε = 9

= 6 .

3

? & c2 = a2 − b2 ! <, % b2 = a2 − c2 = 92 − 62 = 45 .

x2

y2

! ':

+

= 1. O

45

81

43

1.32. # //

# " < '+" @ # 3 # +E 3 ! %, " " > A- + D 3 E + $ > 6 +$ 3 F1 F2 E <9 !- % (, % ) < 3 ! + $ .

" " x2 − y2 = 1 ' 1 & +

a2 b2

. #.

! = # 3 " " @ ! A &

@ ": y = ± b x2 − a2 . a

+ ' > ! # A& ! @ & + <9 > ' 9 @ E

x2 − a2	≈ x2 ≈ x . + < " " !$# @ ":
y = ±kx ,	k =	b	. A ! x → ∞ @ ! A

a

" @ A ' ! > " ' ! "# >, > -

. +. 1.30 ! A

3 $ b D ' ! $+ D $ &" D ! +"# @ ! A.

" !+: ε = c , c2 = a2 + b2 . a

? +. 1.30 a<c, #& " !+ ε > 1.

)

+ 3 " " @ ! A, % ! > ' 3 3 !(4,0)

(5,15/4)

N + ' # 3, % C ' @ ! A, # C + + + + #& > " ', ! + = # + ' 9> 3 " 3 - " ".

		2								2
			4			−		0			= 1							a2 = 42							a = 4
			2					2
			a					b											15		2
												2					2							15		2		3		2
					(15 4)											5			(	4)
	52																	− 1 =									=
				−									= 1		42				b2					4b				4
		2						b		2
a															a = 4					a = 4
																					= 5
															12b = 60					b
! ':							x	2		−		y2		= 1. O
					42							52

44

1.33. ) #

) # "

< '+"

@ # 3

# +E

3

!-

%, " " > + ' " 9

6 + 9 3 F +

" 9 6 + 9 ! "# 9, -

C = 9 $ !-

%. 3 F < '+" 6 &-

+ # ! A, ! "# – -

+ D ! A.

y2 = 2 px

'-

" "

1 & +

#.

p

3

< '+"

! # #

! A.

3

+. 1.31 A

F(p/2;0) < '+" 6 &+ # !-

A, ! "# x = − p

99 + D.

2

+ &< ! =& " @ ! A:

– " % + ! +&< '+" " "#

x = − p

2

,

3 -

" " ! A # < @ ":

y2 = 2 px

99 @ + # 3 % + A+E +.

– " % + ! +&< '+" " "#

y = − p

2 ,

3 -

" " ! A # < @ ": x2 = 2 py

99 @ + # 3 % + .

" ! A: a = c =

p

, #& ε =

c

= 1.

2

a

)

+ 3 " " ! A, " % 99 + ! +&< '+" - " "#: D: x = 1.

N #& ! + ! A # < @ " x = − p 2 . ? " " + # <#: (1 = − p 2 ) ( p = −2). 3 " " !-

A: y2 = 2(−2)x . ! ': y2 = −4x . O

+ &D3 +! = " # C ! # # > &@ @ ! " & # C + A. 1.2:

45

A E" 1.2

! = " # C ! # # > &@ @ ! " &

#

) #

c2 = a2 − b2

c2 = a2 + b2

1) x = − p 2 (y2 = 2 px)

ε < 1

ε > 1

+ # 3

x2

y2

2

2

+

=1

x

−

y

= 1

2

2

a2

b2

a

b

ε =

c

y = ±kx , k =

b

.

2) y = − p 2 (x2 = 2 py)

a

a

+ # 3

1.34.% & ' ( 1.9

& ! ’" & > C3 3, A = 99 + 9# # # & C& @ &!.

?$ 3 " ": ) !+; A) @ ! A; &) ! A, " % -

: !, –

3, " = $; F – 6 &+; – ( $+) ! +';

– # (&") ! +';

ε

– +E + ; y = ±kx – " " + #!

@ !-

A; D – + ! A; 2 – # C6 &+ + '.

1. ) =2 2 , ε =

72 /9; A) k=

2 /2, 2 =12; ) +' + # 9

y !(–45; 15).

2.

) 2 =30, ε =

17 /15;

A) k= 17 /8, 2 =18; ) +' + # 9 y !(4;–10).

3.

) ε =5/6, !(0;–

11);

A) !(

32

; 1), (

8 ; 0); ) D: y=–3.

3

4.

) !(–3; 0), (1;

40 /3);

A) k =

2

, ε =

15 /3; ) D: =4.

3

5.

) =7, F(13; 0);

A) =4, F(–11; 0); ) D: =13.

6.

) =13, F(–5; 0); A) =4,

F(–7; 0);

) D: =– 3/8.

7.

) =2 15 , ε = 7/8; A) k = 5/6, 2 = 12;

) +' + # 9

x !(–2; 3

2 ).

8.

) 2 =50, ε =3/5;

A) k=

29 /14, 2 =30;

) +' + # 9

y !(4; 1).

9.

) ε =2/3, !(–6; 0); A) !(

6 ; 0), (

20

; 2) ; ) D: y = 1.

3

10. ) !(0; –2), (

15 /2; 1);

A) k = 2

10 /9, ε = 11/9; ) D: = 5.

46

11.) =5, F(–10; 0); A) = 9, ε = 4/3; ) D: = 12.

12.) =9, F(7; 0); A) = 6, F(12; 0); ) D: = –1/4.

13.	) =5, ε = 12/13; A) k = 1/3, 2 = 6; ) +' + # 9							y !(–9; 9).
14.	) 2 =22, ε = 10/11; A) k = 11/5, 2 =12; ) +' + # 9 x !(–5; 5).
15.	) ε =3/5 , !(0; 8);			A) !( 6 ; 0), (–2 2 ; 1);			) D: = 9.
16.	) !(–	17	; 1/3), (		21	; 1/2); A) k = 1/2, ε =	5 /2;	) D: =–1.
	3			2

17.) =7, F(5; 0); A) =11, ε =12/11; ) D: = 10.

18.) =6, F(–4; 0); A) =3, F(7; 0); ) D: =–7.

19.	) =2, ε =5 29 /29; A) k=12/13, 2 =26; ) +' + # 9									x !(–5; 15).
20.	) 2 =24, ε =	22 /6 ; A) k=		2	3	, 2 =10; ) +' + # 9				x !(–7; –7).
21.	) ε =7/8, (8; 0); A) !( 6			;–	3		5	), ( 15 ;	6 ); ) D: = 4.
22.	) !(0; 3 ), (	14	/ 1) ;	A) k =				21 /10, ε = 11/10; ) D: = –4.
		3

23.) =4, F(9; 0); A) =5, ε = 7/5; ) D: = 6.

24.) =4, F(3; 0); A) =2 10 , F(–11; 0); ) D: = –2.

25.	) =	15 , ε =	10		;	A) k=3/4, 2 =16 ;		) +' + # 9 x !(4;–8).
			25
26.	) 2 =22, ε =		57 /11; A) k=2/3, 2 =10				13 ; ) +' + # 9 x !(27; 9).
27.	) ε =	21 /5, !(–5; 0);				A) !( 80 ; 3),	(4 6 ; 3 2 ); ) D: =1.
28.	) !(3; 0), (2;		5 /3);			A) k=3/4, ε =5/4;		) D: =–2.
29.	) =2; F(4 2 ; 0);			A)		=7, ε = 85 /7 ;		) D: =5.

30.) =15, F(–10; 0); A) =13, ε =14/13; ) D: =–4.

31.) =7, F(–3; 0); A) =3, F(9; 0); ) D: =–5.

1.35. ) 0 %. . %

" " F(x, , z)= 0 3 < & ! + Oxyz " & ! > D, A @ # 3 # +E 3, " > x, , z ' "D ' E' #& " D. / " " < '+" " "# ! >, x, , z – ! 3 # -#.

47

> "# &@ @ ! " & D '+" ! >, " # C 3 " "# &@ @ + &! " % > ! "# & > . !-

, +6 x2 + y2 + z2 = R2 – ! > " &@ @ ! " & (+. 1.32). + &< ! ! > ' &@ @ ! " &:

1./ ' ! > &@ @ ! " &.

2.A 9.

3.B 3 $ E 3 ! > &@ @ ! " &.

1.35.1. * & 0 %. . %

	#							! E ' > ! > ' &@ @ ! " &:
	) !+ 9 ( $+$ A &"$);
	A) ! C$ @ ! A 9;
	) ! C$ @ ! A 9.
	+ # D ' <$ E + # 9.
	/ ( $+ #) < '+" ! > ", " & # C 3 -
" "#		x2	+	y2	+		z	2	=1 – E 3 " " !+ 9.
		a2		b2		c2
											x2 + y2 + z2 = a2 .
	% a = b = c , !+ 9 < +6 D &+& a :
	'$								!+ 9 # C	C	"@& "# +6
x2 + y2 + z2 = 1					&			! "# >		+$ 6 E < # "@& "
a,b,c . $+, " % 3 #										x0 , y0 , z0 = E$ +6,

3 ax0 ,by0 ,cz0 = !+ 9.

3 < E # + # 9 !+ 9, @ 3 # -

" "# (E !+ 9).

" " !+ 9 # C + C& # # ! . -

!+ 9 ! % D z = h 3 < '+" + + # D " '

x2

+

y2

+

z2

= 1,

x

2

+

y2

=

1 −

h2

,

A

b2

c2

b2

c2

a2

a

2

z = h

z = h.

h

> c " ! & – ! C " # C;

h = ±c " ! & –

3 C (0,0, ±c);

h

< c " ! & – !+ ! +"#:

a = a 1 −

h2

, b = b 1 −

h2

.

1

c

2

1

c2

+ !+ # D ' $A '= 3 " ! h = 0 : a1 = a , b1 = b .

!+ 9 # < ! % + # 9. " !+ 9, @ 3 # " "#, E ! % x = 0, y = 0, z = 0 . 9 ! & !+ 9 ! % #

+ # 9 D '+" $@ @ # ! #.

48

+. 1.32. 6

+. 1.33. !+ 9

" " @ > ! # D ' @ ":

2

y

2

2

z

2

2

z

2

x

+

= 1,

x

+

= 1,

y

+

= 1,

b2

c2

c2

a2

a2

b2

z > 0.

y = 0.

x = 0.

	? D3 @ ! , # C ! A& & !+ 9 (+. 1.33.). -
=	@ > ! D '+" = #	!+ 9:				A(±a,0,0),
B(0, ±b,0), C (0,0, ±c).
	= # C& ' A& $ C " 3 ! & !+ 9 $@
+"#	+ # 9 (+"# ). !, !	x = 0, y = 0				C&<#
z = ±c (	3 ! & ++D Oz).		x2	+ y2		z2
	% a = b , " " !+ 9 !$# < @ ":				+		= 1.
				a2		c2

E' #& ! & ! !+ 9 ! % # z = h < # E - # + Oz. $ !+ 9 < '+" !+ 9 # A ". #& !-

& !+ 9 # C A& & $ A "# 9

x2

+

z2

= 1, y = 0

a2

c2

+ Oz.

x2

y2

z2

" "

+

+

= −1 3 < &"$ !+ 9 – ! C D #-

a2

b2

c2

C & 3.

5 . # / < '+" ! > ", " & # C -

3 " "#

x2

+

y2

−

z2

= 1 (+. 1.34).

a2

b2

c2

! C @ @ ! A 9 ! % # z = h , ! ! &- " # + Oz, < !+ #; ! z = 0 !+ # < $# = ! + - < '+" @ # ! #. ! C @ @ ! A 9 ! %-

49

#, ! ! & " # = > > +$, < @ ! A # A - ! D '+" ! & ! "# >.

!, ! $@ ! % D y = h 3 < '+" " "#

x2

−

z2

=

1 −

h2

,

c2

b2

a2

y = h.

# >

h

< b

h

> b C&<# '" @ ! A, ! h = ±b – ! & ! "-

! ! C @ @ ! A 9 3 D '+" "-

2

y

2

x

2

z

2

2

z

2

"#

x

+

= 1,

−

= 1,

y

−

= 1,

b2

c2

c2

a2

a2

b2

z = 0.

y = 0.

x = 0.

A(±a,0,0),

! C$ @ ! A 9 # < ! $+ > =

B(0, ±b,0)

&"

=

C (0,0, ±ic).

+ ’" "D '+" !

x = 0, y = 0 .

" 3 " z ! &D ', % +' Oz ! <

! > D.

a = b

C&<#

! C$ @ ! A 9

A "

x2 + y2

−

z

2

= 1.

a2

c

2

5 . # / < '+" ! > ", " 3 < '+"

" "#

x2

−

y2

−

z2

=

1 +. 1.35). " &3 + A C " ! % xOz

a2

c2

b2

(

+ % ! % D + ". +' Oy +! "# ! % & + ", % A + + # = +" ! D.

+. 1.34. ! C$	+. 1.35. ! C$
@ ! A 9	@ ! A 9

50