Chuyên đề 2 mặt nón, mặt trụ, mặt cầu

Trần Só Tùng Khối tròn xoay
I. Mặt cầu – Khối cầu:
1. Đònh nghóa
• Mặt cầu:
{ }
S O R M OM R   = =
• Khối cầu:
{ }
V O R M OM R   = ≤
2. Vò trí tương đối giữa mặt cầu và mặt phẳng

• !d < R"# $ !%&'()*) +,-+.
/-01
2 2
r R d= −

• !d = R" !2-(- !& 3.(P) đgl tiếp diện của (S)
• !d > R"0+-& 3
Khi d = 0 thì (P) đi qua tâm O và đgl mặt phẳng kính, đường tròn giao tuyến có bán
kính bằng R đgl đường tròn lớn.
3. Vò trí tương đối giữa mặt cầu và đường thẳng
&'(∆∆
• !d < R"∆#& 3+/ 4
• !d = R"∆ !2-(-
∆
đgl tiếp tuyến của (S)
• !d > R"∆0+-& 3
4. Mặt cầu ngoại tiếp – nội tiếp
Mặt cầu ngoại tiếp Mặt cầu nội tiếp
Hình đa diện
5!6-&76"& 4

& *) +
5!6-6"& 4
&  !2-(-
Hình trụ
.&'()&-$6")
*) +
8 !2-(--&-$
&'(96")
Hình nón
8&:&7&'()
&-$6"-
8 !2-(-&-$
&'(96"-
5. Xác đònh tâm mặt cầu ngoại tiếp khối đa diện
•-;< !(n – 2)&76& 4"&7%'(-4-+"
+6%)& 36&!&7&-
•-=<> 32-&?+6 !"-
@A-&?)∆6&-$(
∆
là đường thẳng vuông góc với đáy tại tâm
đường tròn ngoại tiếp đa giác đáy).
@A-&?))'64/ +
@& 36∆%+6 !"-
BBC 41@5 31
Cầu Trụ Nón
C 41
2
4S R
π
=

2
xq
S Rh
π
=
2
tp xq đáy
S S S= +
xq
S Rl
π
=
tp xq đáy
S S S= +
5 31
3
4
3
V R
π
=
2
V R h
π
=
2
1
3
V R h
π

=
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CHƯƠNG II
KHỐI TRÒN XOAY
CHƯƠNG II
KHỐI TRÒN XOAY
Khối tròn xoay Trần Só Tùng
VẤN ĐỀ 1: Mặt cầu – Khối cầu
Bài 1. "-DE-&-$DE%-+E
)(ABCSA ⊥

%)& 36'-<DE$)/!& 3
D,E,,*) ++/-01
2
SC
R =

/DE
2aAB =
51/-01-) +
Bài 2. 5),&'(4& 3D84-2D$
&4:D,#E5) +&'(:D+-(-%!$& 3
.F%-" !+-6D) +E
'-D,E,,.,F44
/51/-01) +,/ !DE=,DG,
·
0
0BAC 6=

Bài 3. "-DEC-&-$DEC%"+,

)(ABCDSA ⊥

3aSA =
%+"+DECF%" !6E) +
-/& 3,D,F"&E'(-4-+$)H
& 3,C,D,FE*) +&'(01E
/A-&?+/-01-) +
Bài 4. 4& 3D,/ !D=ID0 64 !$ ! !
2-(-EJ:D0 64-$ !#C,/ !
3aCD =

51DE
/5106-'& !&'(C
Bài 5. "--& DE,-&-$/*-(/(6/ +
&-$/*KL
L
%+6-DE5)-D'&'(
))'6D,#F
51,D
/'-
SMK SOA
∆ ∆:
(-8%)& 36D$)F
'-"-FDE%"-& 9$)<FDFEMF
A-&?+/-01 !"-DE
Bài 6. "-DE/ !)*-4/-01 !2-(--
6"-+B6*) +&'(.6"-
'-)*DE%"-& 
/51 6"-,/ !)*
3RIS =

Bài 7. '- 4& DEC-%
A-&?+/-01 !'- 4
/51 41 310!&-
Bài 8. 4"-'--& -&-$%,/ +((-&-$4
-KL
L

A-&?+/-01 !"-
/51 41 310!&-
Bài 9. "-'--& DEC-!6-& /*A-&?+
/-016&:H& 3,D,E,,C
Trang 16
Trần Só Tùng Khối tròn xoay
Bài 10. -DE-&4/%;G,;N,;O84+,/-01
O !2-(-/6-DE- !& 3*) +/&-
5106-'+(-'--
Bài 11. ."-DE-&'(D,&-$DE%-& 51
/-01 !"-
Bài 12. "-'-& DEC-&-$/*-(/(6/ +
&-$/*KL
L
A-&?+/-01 !"-
Bài 13. ."-'--& DEC-&-$&'(%+6
DEC.%)& 36E>'(+-)6-.#B
'-)*B%+4 !"-51/-01$
Bài 14. "-DE-D⊥DE-DE+ED.,DF
%%'(%-&'(6--DED
'-)*H& 3D,E,,.,F(6) +4
/DE;L,E=NA-&?+1/-01&-
Bài 15. "-DEC-DEC%"+/*,D

7a
D⊥
DEC84:D+-(-,#E,,C%%'(
.,8,F
'-)*/6$& 3D,E,,C,.,8,F(6) +4
/A-&?+1/-01&-
VẤN ĐỀ 2: Mặt trụ – Hình trụ – Khối trụ
Bài 1. ")--&-$%")+′,/-01&-$/*=
5) +&'()&-$+%!$& 3D,E9DE=E !)* 31
'- 4′DE/*P
G
51 ") 310!)
Bài 2. ")--&-$%")+′,/-01&-$/*=
5) +&'()&-$+%!$& 3D9D′((-&-$4-
0
60
51 ") 310!)
Bài 3. ")--&-$%")+′,/-01&-$/* 
/*5) +&'()&-$+%!$& 3D,) +&'()&-$+′%!$
& 3E9DE=51 3160!'- 4′DE
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Trang 17
Khối tròn xoay Trần Só Tùng
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VẤN ĐỀ 1: Mặt nón – Hình nón – Khối nón
Bài 1. "%H)'--& DECD′E′′C′-&-$/*, =
E !)*′%+6D′E′′C′%&'()4 !&-$DEC51 3
10!--&7′&-$
Bài 2. "%H)-& DED′E′′-&-$/* =
E !)*′%+6D′E′′%&'()4 !&-$DE51 31
0!--&7′&-$
Trang 18
Tran Sú Tuứng Khoỏi troứn xoay
Baứi 3. "-'--& DEC-&-$/*,/ +((-&-$
4-
0
60
%&'() !&-$DEC51 310!--
&7&-$
Baứi 4. 5)0+-B8+B,-B8/*GL
L
B8
F:$-B8:-+B"&'(!0-8B
4"-)2$
51 412:6"-)2$
/51 3160!-)2$
Baứi 5. 5 ! 4:)64"-%4-++--
+/*
51 412: 416"-
/51 3160!-'('-
84 ! 4:&7(-&-$4-KL

L
51 416 ! 4
$
Baứi 6. "-&7,&'(,DE%& 34&'()&-$9
06-'& 3& !DE/*
ã
0
0SAO 3S=
,
ã
0
0SAB=6
51&4
&'(96"-
Baứi 7. 5 ! 4:)640!-%4-++-$
/*51 310!- 412:6"-&J
Baứi 8. "%4'(DECDQEQQCQ51 412:6
"--&7%+6"+DEC&-$%")4 !"
+DQEQQCQ
Baứi 9. #4"-/*4&:)6-,&'( ! 4%
4-& =51 412:, 416"
3160!-
Baứi 10. "--& DE-/ +/*-'J-/ +
&-$%

84"-&7-&'()&-$4 !-& DE,
.J$1 412:6"-$


Baứi 11. "-'--& DEC-

ã
SAB

=


TNO
L

51 412:6"-&7-&'()&-$ !"
+DEC
Baứi 12. 84"--&4&'(9/*;-'J&'(9&-$%


5" 412: 3160!-
/B%& 3) +&'(6"-9
( )
10 <<= kk
SO
SI
51
416 ! 4:B+-(-)
Trang 19
Khoỏi troứn xoay Tran Sú Tuứng
Baứi 1. 4'- 4& -%
A-&?+/-01 !'- 4
/51 41 310!'('-
Baứi 2. 4"-'--& -&-$%,/ +((-&-$4
-
0

60

A-&?+/-01 !"-
/51 41 310!'('-
Baứi 3. ."-'--& DEC-&-$,-'J/ +&-$%
51/-01- !4 !"-
/51-)?6


& 3-$-+)
Baứi 4. '- 4DEC,/ !DEEDEC,DC/.DC
EC+-(-
'--DC+
/51 41 !'- 4DEC
Baứi 5. "+/-01&'(0184+-(-
#" 4&'()+.DE%-& 4 !)
&'()$>.x (0 < x < 2R).
51-6'- 4DE R, x.
/A-&?x& 3DE%'- 4& ,0&-1 316'- 4'-
)*-&'(D,E,&+4+-(-
Baứi 6. 5),"+DEC(-DE=,ECCD
5) +'6&'(D2+-(-%!$4& +&484
:D+-(-E,#E,,C%%'(,I,
'-)*/6$& 3D,E,,C,,I,%+44!&?
1 416&-
/D
3a
51 416'--DI
Baứi 7. 4&BU- 5) +&'(+-(-BUB
%!$& 3D,D&!2'-:BBDBD5) +&'(+-(-BU

U0+99(-DD%!$& 3E,E&!2'-:UUEUE/
'-)*+6 !'- 4DDEE*) +&'(
BU
/A-&?+1/-016 !'- 4DDEE ,/,
Baứi 8. '- 4DEC(-DED,E/.ECDE
+-(-
ã
0
90BDC =
A-&?+/-01 !'-
4DEC
Baứi 9. "/-015'4& 3/!0$) +,'/-$ !
/*,#D,E,9<
ã
ã
ã
ASB ASC =BSC

= =
51 31R
6'- 4DE


Baứi 10. '- 4DE-DDE,D,DE/,DA-&?+1
/-01 !'- 4)-)'((9<
Trang 20
ễN TP KHI TRềN XOAY
ễN TP KHI TRềN XOAY
Tran Sú Tuứng Khoỏi troứn xoay

ã
0
90BAC =
/
ã
0
60BAC =
,/
ã
0
120BAC =
,/
Baứi 11. "%H)-& DEDQEQQ-!6-& /*A-
&?+,/-011 41 !"%H)&J
Baứi 12. 84")-/-01&-$- ! 4:)%4"+
51
2:


6")
/51R0!%H)'--& 4 !)0!)&J
Baứi 13. 84")-/-01&-$&'(
3R
DE%=& 3) +=
&'()&-$9-(/(6DE)6")%
0
30

51 416 ! 4:DE99(-)6")
/51

2:


6")
51 310!)'('-
Baứi 14. E +)"))2$-4"+DEC4 !=&7
% + !D,E*) +&'()&-$'-;6"),=&7%*) +
&'()&-$'-=6")8'-"+(-&-$")
4-
0
45
51 412: 316")&-
Baứi 15. 5 ! 4:)64"-%4-++--
+/*
51 412: 416"-
/51 310!-'('-
Baứi 16. "--&'(h/-01&-$R8%& 3) +&
,&8x (0 < x < h).
51 41 ! 4+-(-)8
/51 31R60!-&7&-$ R, h vaứ xA-&?x9R
&-)?%(-!
Baứi 17. 84"-&7- .h&'(9/*&'(01&-$
84"-+%)& 36&'(. !2-(-&-$"-
A-&?$ !6-
/51 416-*)
5199-(- 416-
Baứi 18. "-)2$&75)&-$6"-&--"+DEC
4 !,/*E !)*
ã
0 0

2 0 45ASB ,

= < <
51 310!-
412:6"-
Baứi 19. "--/-01&-$/*-(6&7%=

5)"--
4")4 !51/-01&-$ 6"),/ !)* ! 4
:)6")%4"+
Baứi 20. "--/-01&-$,-'J&'(9&-$6"-%


8499(-&-$6"-,-&-$"-406,
#"- &'()51/-01&'() ,



Trang 21
Khối tròn xoay Trần Só Tùng
Bài 1. "--DE-&-$DE%-& ,D⊥DE
D8%4& 3$&3) +DE>
·
D8
α,.+-
(-&'(8
5":$J1& 3.$)-)?%(-!6 31'- 4D.
/.DB⊥,DF⊥.51&4F,DF 31'- 4DFB
HD: a) Q tích điểm H là một cung tròn. MaxV
SAHC

=
3
12
a

b) AK =
2
1
a9
9
α
α
+
,

SK =
2
1
a
9
α
+
, V =
3
2
2
24 1
a 9
 9 
α

α
+
Bài 2. ∆DE+D-DED-
·
ED
=α5) +&'(:
D+-(-DE,%!$& 39D=B%)
& 36E.D.⊥B
'-D.⊥E51&4D. ,α
/F%4& 3$&3) +&DB,&
DF
DB
28:F+
-(-DB#-DE,D,,E%%'(8,,,I5'--8I%
""V51 41'--$
HD: a) AH =
2
2
4
a9
9
α
α
+
b) S
MNPQ
=
( )
2
4 1a x x a@ 9

Bài 3. '- 4DEC-DEC=2
 
 
 
 
=
LW2W
=
DDCEEC;
BU%%'(%)& 36-DEC
'-DE⊥CBU%&+-6&'(DE
C
/51 31'- 4DEC 25"2& 3 31$%(-!1-)?%(-
!&-
HD: b) V =
2 2
2 1 2
3
x x−
; MaxV =
2
9 3
khi x =
3
3
Bài 4. 5),"+DEC,-+%5) +-'6
&'(D2,$+-(-(6 41&!(-%!$%%'(
& 38,>D82,$
51&485'&-'-)*& 0 4&6& 3∆8+
%<

2
2xy a=

/69'68,$&39∆8+51 31'- 4EC8
A-&?2,$& 3 31'- 4$/*
G

N

Trang 22
ƠN TẬP TỔNG HỢP HÌNH HỌC KHƠNG GIAN
ƠN TẬP TỔNG HỢP HÌNH HỌC KHƠNG GIAN
Trần Só Tùng Khối tròn xoay
HD: a) MN =
2 2
2a x y + −
b) V =
3
6
a
x y +
, (x, y) =
2
a
a
 
 ÷
 
hoặc
2

a
a
 
 ÷
 
.
Bài 5. 5),"+DEC%& 36=
&'( -6"+DEC5) +&'(2+-%!$& 3
α%-/(6/ +&-$6"-DEC
51 31 416"-DEC α
/A-&?&'(+-6DC51&4&'(+-
&- α
HD: a) V =
3
6
a

α
, S
tp
=
2
1
1a
9
α
 
+
 ÷
 

b) d =
a 
9
α
α
Bài 6. 5) +'6&'()&'(01DE=%!$4& 3$$-C'.+
-(-DE.4&DEB%)& 36.5) +'6&'(
B+-(-DEB%!$& 39-
·
DE
XL


'--.%-& 
/>D.51 31R6'- 4DE 
HD: b) V =
( )
3
2
Rh 2R h@

Bài 7. "+DEC=5) +&'(:)& 3B6
DE+-(-DEC%!$& 3Y9BY8%& 3$
&3) +DE,Y.⊥8>E82
'-& 3.&4) +4&'()51&4B.
/U%)& 36&Y51&4U8"-)?6!6U8
HD: a) IH =
2 2
2
4

a x a
a x
−
+
b) JM =
2
2
5
2 4
a a
x
 
− +
 ÷
 
,
MinJM =
5
2
a
khi x =
2
a
Bài 8. "4'J4DECDZEZZCZ& 38) +DC8
DZE8#&'( -DZ6"4& 3.
'-)*08$&3) +DC"&'(8.#&'(
DZE4& 3!&?
/51$69! 3160!& 4/(6DZE8#"4)
)'((8%)& 36DC
69'6DDZDE8E+-(-D'-)*DZE8

+-(-DZ& 3.%)'+6-DZE8
HD: a) MH cắt A
′
B tại trung điểm I của A
′
B. b)
1
2
1
11
V
V
=
Bài 9. "+DEC/*B%)& 3DEIB'&'(+
-(-DEC) +&-%!$& 39=B
G

'-)*-DC%-+
/51 310!-DC)9$)06-'& !DC
HD: b) V =
3
3
12
a
, d =
3
2
a
Bài 10. "4'J4DECDQEQQCQ-DE,DC=,DDQ
5106-'J&'(DCQEQ

Trang 23
Khối tròn xoay Trần Só Tùng
/8%& 3)&DC $69!
3
AM
MD
=
.J$106-'
& 38& !DEQ
51 31'- 4DEQCQ
HD: a) d(AD
′
, B
′
C) = a b) d(M, (AB
′
C)) =
2
a
c) V =
3
2
3
a
Bài 11. 5),4"+DEC-/*%4& 3
/!0$*) +&'(D+-(-D
51  310! !-DEC0D=
/8,%%'(%& 3&4) +-E,C8∈E,∈C&
8,5"4/ 3'-% + 4'J& 3-8D
D(-4-NO°

HD: a) V =
G
 Kπ
b)
( )
0
2
2a 2 m n a mn@ + + =
Bài 12. "- DEC -&-$ DEC %"+, 
( )SA ABCD⊥

2SA a=
5) +DC%!$& 38$&3>-
·
α
=ACM
.
SN CM
⊥

'-%+44&'()!&?1 31'- 4D
 
a

α

/.
AH SC
⊥
,

AK SN
⊥
'-)*
( )SC AHK⊥
1&4&.F
HD: a) N thuộc đường tròn đường kính AC cố đònh, V =
3
2
2
6
a
9
α
b) HK =
cos
2
1 sin
α
α
+
a
Bài 13. "-DE--/ +D,E,&+4+->D
,E/,%)+6-DE
51&4& ,/,
/84$$-&:#&DE8#&D
'-)*
3
AB AC
AM AN
+ =


'-)*&:-& 3,D,E,-+4
51 310!& 4D8 ,/,09
9(-E
HD: a) SG =
1
2 2 2
3
+ +a b c
b) V =
1
9
abc
Bài 14. "+DEC%& 3&'( -5) +'6
&'(2+-(-'-"+,%!$& 39-
·
60= °SCB

5106-'J&'(EC
/
α
%'-E+-(-DC51 41
 ! 4/(6
α
"-DEC
HD: a) d(BC, SD) =
6
3
a
b) S =

2
6
4
a
Bài 15. "+DEC-/*5) +DC%!$& 389D8
x (0
≤
x
≤
a)5) +'6&'(D2+-(-DEC& 3D,
Trang 24
Tran Sú Tuứng Khoỏi troứn xoay
%!$& 39D$$TL
'-)*DEE
/5106-'& 38& !D
51 310!-DE8 ,$2
E !)*2
=
M$
=

=
5"-)?%(-!6 310!-DE8
HD: b) d(M, (SAC)) =
2
2
x
c) V =
1
( )

6
ya a x+
d) MaxV =
3
3
8
a
khi x =
2
a
Baứi 16. "-DE-&-$DE%-+D
ã
0
30ABC =
E%
-& 8/ +DE+-(-&-$DE8%)& 3E
'-D8%&+-6ED519-'J=
DDE
/51 316"-DE
HD: a)
ã
1
3
SAB9 =
b) V =
3
2
24
a
Baứi 17. "-DEC-&-$DEC%",-

à
0
120A =
,ECTL
/ +D+-(-&-$-'JE&-$/*KL
L
84
&:EC+-(-5179! 31'J
6"-)0#"-
HD:
1
2
1
12
V
V
=
Baứi 18. "4&'-DECDQEQQCQ--DEDC,DDQ
2
3a

-
ã
0
60BAD =
8%%'(%)& 36-DQCQDQEQ
'-)*D+-(-EC851 310!-
DEC8
HD: V =
3

3
16
a
Baứi 19. "-DEC-&-$DEC%"'J4(-DE,DC=,
D+-(-&-$,E(-&-$4-KL

5) +
D%!$& 389D8
3
3a
8E8#C& 351
310!-E8
HD: V =
27
310
3
a
Baứi 20. "-DEC-&-$DEC%",-
ã
0
60BAD =
,D
+-(-DEC,DQ%)& 368
&:DQ99(-EC,#-E,C6"-%%'(
EQ,CQ51 3160!-DEQQCQ
HD: V =
18
3
3
a

Trang 25
Khối tròn xoay Trần Só Tùng
Chân thành cảm ơn các bạn đồng nghiệp và các em học sinh đã đọc tập tài liệu này.
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