重难点23 与圆有关的最值与范围问题【十大题型】
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1 !"#$%&
é 1 "23-24 á-·êëìí·îïðñ% v𝑃(𝑚,𝑛),𝐶:(𝑥−1)2 + (𝑦−1)2 = 1-ò¦*I¤
𝑚+𝑛
𝑚+1.
!" %
A 3
3
B−3
3
C1 +
3
3
D1−3
3
}\ 1-1 "2024·óô·õö÷ø% v*𝑃(𝑥,𝑦)z,(𝑥−1)2 + (𝑦−1)2 = 3-{|I¤
4−𝑦
𝑥−3. !
" %
A−6−30
B6 +
30
C−6 +
30
D6−30
}\ 1-2 "2024·ùúûü·âõ% v𝑃(𝑥0,𝑦0)J,𝐶:𝑥2 + 𝑦2−2𝑥−2𝑦+ 1 = 0-ò¦*I¤
𝑦0+1
𝑥0−3.
!" %
A−2
B−
1
2
C−4−7
3
D−4+ 7
3
}\ 1-3 "2024·ýÃôP·âõ% v𝑃(𝑚,𝑛),𝐶
(𝑥−1)2 + (𝑦−1)2 = 1-ò¦*I¤
𝑛−1
𝑚+1. !
.
2 '( !"#$%&
é 2 "23-24 â-·óô·îïðñ% v*𝑃(𝑥,𝑦)J,𝐶
(𝑥−𝑎)2 + 𝑦2 = 3(𝑎> 0)-.|*Iþ,𝐶
ÿ1*𝐴(1, 2)I¤𝑦−𝑥. !? !Ā¡" %
A4
B2 6
C−4
D−2 6
}\ 2-1 "24-25 á-·āĂ·ăÉĄą%ÒĆ
𝑥,𝑦ÕÖ\𝑥2 + 𝑦2 +4𝑥−2𝑦−4 = 0IćĈ𝑥2 + 𝑦2.
!J ²2𝑥−𝑦. !J
}\ 2-2 "23-24 á-·ĉĊċČ·îïðñ% v𝑥I𝑦JĆ
Ič(𝑥−1)2 + (𝑦−2)2 = 4
(1)u3𝑥+ 4𝑦. !²
(2)u
𝑦
𝑥.Ď!#$²
(3)u 𝑥2 + 𝑦2. !
}\ 2-3 "2024 â·āĂ·ðñ% vĆ
xIy ÕÖ[ x2ďy2Đ4xď1đ0.u
(1)
𝑦
𝑥. !¡ !²
(2)yďx . !¡ !²
(3)x2ďy2 . !¡ !
3 )*+,-*. !"#$%
é 3 "2024·ùúĒē·âõ% v,𝐶:(𝑥−𝑎)2 + (𝑦−𝑏)2 = 1ÿ1*𝐴(3,4)I¤Ĕ,+ĕ*.xy.
!" %
A4
B5
C6
D7
}\ 3-1 "23-24 âĖ·ėĘęô·Ě ě% v𝑃J,𝑂:𝑥2 + 𝑦2 = 9-.|*I*𝑄ÕÖ𝑃𝑄= (3,−4)I
*𝐴(1,1)I¤|𝐴𝑄|. !" %
A8
B9
C 29 +3
D 30 +3
}\ 3-2 "2024·āĂ·õö÷ø%𝑀*J,𝐶:(𝑥+ 2)2 + 𝑦2 = 1-ò¦*I𝐴𝐵,𝐶1:(𝑥−2)2 + 𝑦2 = 3
.3Ič|𝐴𝐵| = 2 2I𝑁𝐴𝐵.~*I¤|𝑀𝑁|. !" %
A1
B2
C3
D47
}\ 3-3 "2024·Ĝēĝė·âõ% v,𝑂:𝑥2 + 𝑦2 = 16I*𝐹−2, 1
2 +
19 I*𝐸J𝑙:2𝑥−𝑦+ 16 = 0-
.|*I1𝐸Ą,𝑂.5(I5*ºÎ𝐴I𝐵I'(𝐴𝐵?𝐸𝑂ĞM*𝑀I¤|𝑀𝐹|. !" %
A
3
2
B3 5
2
C5 5
2
D3 19
2
4 ,-*+)'("/0%-. !"#$%
é 4 "2024·óëğĠ·õö÷ø% v MIN J, C
𝑥2 + 𝑦2−2𝑦−3 = 0-.Û°*Ič|𝑀𝑁| = 2 2IP
𝑀𝑁.~*IQ '(𝑙
𝑥−𝑦−3 = 0-.*I¤|𝑃𝑄|. !" %
A2 2
B 2
C2−2
D 2−1
}\ 4-1 "2024·ġĢģė·áõ% v'(𝑙:𝑥−𝑦−2 = 0I*𝐶z,(𝑥−1)2 + 𝑦2 = 2-{|IćĈ*𝐶+
'(𝑙.xy. !" %
A
3
2 2 +1
B
5
2 2
C
3
2 2
D 2
2
}\ 4-2 "2024·óë·áõ% v𝐴(𝑥1,𝑦1)I𝐵(𝑥2,𝑦2)J,𝑥2 + 𝑦2 = 9-.Û°|*Ič𝑥1𝑥2 + 𝑦1𝑦2 = −
9
2Iþ*𝑀ÕÖ𝐴𝑀= 2𝑀𝐵I*𝑃z'(𝑥+
3𝑦−4 3 = 0-I¤|𝑀𝑃|. !" %
A4 3
B3 3
C2 3
D 3
}\ 4-3 "2024·êôĤĥ·áõ% v*𝐴(𝑥1,𝑦1),𝐵(𝑥2,𝑦2)J,𝑥2 + 𝑦2 = 16-.Û*Iþ∠𝐴𝑂𝐵=
π
2I¤
|𝑥1 + 𝑦1−2| + |𝑥2 + 𝑦2−2|. !" %
A16
B12
C8
D4
5 1,2)*.34 !"#$%&
é 5 "23-24 á-·Ħħ·Ĩĩ% v,.[𝑥2 + 𝑦2−8𝑥= 0I¤¶,~1*𝑃(2,1). µ3.4
" %
A 10
B 11
C2 10
D2 11
}\ 5-1 "2024·ùúúĪ·õö÷ø% v'(𝑙:𝑡𝑥+ 𝑦−2𝑡−3 = 0(𝑡∈R)?,𝐶:(𝑥−1)2 + 𝑦2 = 16RĞ
M𝐴,𝐵Û*I¤34|𝐴𝐵|.Ď!#$J" %
A[2 3,8]
B[4 3,8]
C(4 3,8)
D[4,4 3]
}\ 5-2 "23-24 á-·īĘĬĭ·Ĩĩ% v'(𝑙
𝑚𝑥−𝑦−3𝑚+ 1 = 0Į1*𝑃I1*𝑃Ą'(?,
C
(𝑥−1)2 + (𝑦−2)2 = 25RĞM AIB Û*I¤|𝐴𝐵|. !" %
A4 5
B2
C4
D2 5
}\ 5-3 "2024·ċúįİ·áõ% v'(𝑙:(𝑚+ 𝑛)𝑥+ (𝑚−𝑛)𝑦−2𝑚= 0(𝑚𝑛≠0).,𝐶:(𝑥−2)2 + (𝑦−2)2
= 8I¤" %
Al 1)*(1,−1)
Bl ? C )RĞ
Cþ l Pº C .74I¤𝑚= 1
Dl ı C ©. µ3.46 4
6 ,.5(46 !"#$%&
é 6 "2024·āĂ·õö÷ø% v P '(𝑙:𝑥−𝑦+ 1 = 0-*I1* P Ą,𝐶:(𝑥−1)2 + 𝑦2 = 1.£
5(I5* AI¤|𝑃𝐴|. !" %
A1
B 2
C 3
D2
}\ 6-1 "2024·IJ·áõ%A'(𝑥−𝑦+ 2 = 0-.*ij,𝑥2 + 𝑦2−4𝑥−4𝑦+ 7 = 0Ĵ5(I¤5(4.
!" %
A 2
2
B1
C 2
4
D 2
2 −1
}\ 6-2 "2024·ĜēĵĶ·áõ% v*𝑃J'(𝑥+ 𝑦+ 3 = 0-|*I1*𝑃Ą,𝐶:(𝑥+ 1)2 + 𝑦2 = 1
.£5(I5*𝐴I¤(ï𝑃𝐴46. !"ķķ%
A2 3
B2 2
C 2
D1
}\ 6-3 "2024·êë·õö÷ø% v*𝑃'(𝑙:3𝑥−4𝑦+ 12 = 0-.*I1*𝑃Ą,𝐶:(𝑥−3)2 + (𝑦−2)2
= 1.5(𝑃𝑀I5*𝑀I¤5(4|𝑃𝑀|. !" %
A
12
5
B
13
5
C 170
5
D 194
5
7 7489 !"#$%&
é 7 "2024·-ĭĸĹ·áõ%'(𝑙ÿ1)*𝑃(2,1)Ič?𝑥ĺàĺ=𝑦ĺàĺºÎRĞM𝐴I𝐵Û*I𝑂
;<ĕ*I|,𝑀z △𝑂𝐴𝐵.¢ĻIč?'(𝑙³Û;<ĺ.àĺļR5I¤ △𝑂𝐴𝐵74. !J
" %
A3
B5
C10
D12
}\ 7-1 "2024·ėúĽľ·õ% v,𝑄:(𝑥−4)2 + (𝑦−2)2 = 4I*𝑃'(𝑥+ 𝑦+ 2 = 0-.|*IĿ
𝑃𝑄'.,?,𝑄RĞM𝐴,𝐵Û*I¤Ĝŀ0𝑃𝐴𝑄𝐵89. !" %
A2 7
B4 7
C2
D4
}\ 7-2 "2024 â·āĂ·ðñ%Ú P '(𝑥−𝑦= 0-.|*IPAIPB , C
(𝑥−2)2 + 𝑦2 = 1
.Û£5(I5*ºÎ AIBI¤Ĝŀ0𝐴𝑃𝐵𝐶.74. !" %
A3
B2 +
3
C4
D2 + 2 3
}\ 7-3 "2024·āĂ·õö÷ø% v𝐴(−3,0)I𝐵(0,3)IÚ𝐶J,𝑀:𝑥2 + 𝑦2−2𝑥−3 = 0-|*I¤ △𝐴𝐵𝐶
89. !? !ĀŁM" %
A12
B6 2
C6
D3 2
8
:9 !"#$%&
é 8 "2024·ùúĪł·õö÷ø%zP8'>;<T~IW(𝑦= 𝑥2−4𝑥+ 1?;<ĺ.Ğ*z,𝐶-I
𝐴𝐵,𝐶.'I*𝑃J'(3𝑥+ 4𝑦+ 10 = 0-ò¦*²¤𝑃𝐴⋅𝑃𝐵. !" %
A4
B12
C16
D18
}\ 8-1 "2024·āĂ·õö÷ø% v,𝑂J,ĕ*.Ńń,I𝐴,𝐵J,𝑂-ò¦Û°ÑÝ.*I𝑀
(2,0)I¤|𝑀𝐴+ 𝑀𝐵|.Ď!#$" %
A(1,2)
B(1,3)
C(2,4)
D(2,6)
}\ 8-2 "2024·óôĚŅ·áõ% vŀ △𝐴𝐵𝐶.ŀ4 3IP △𝐴𝐵𝐶´zP82.|*Ič|𝑃𝐴
| = 1I¤𝑃𝐵⋅𝑃𝐶.Ď!#$J" %
A −3
2 , 9
2
B −1
2 , 11
2
C[1,4]
D[1,7]
}\ 8-3 "2024·óëņė·áõ% v,𝐶
𝑥2 + (𝑦−3)2 = 4I1*(0,4).'(𝑙?𝑥ĺĞM*𝑃I?,𝐶
ĞM𝐴I𝐵Û*I¤𝐶𝑃⋅𝐶𝐴+ 𝐶𝐵.Ď!#$J" %
A[0,1]
B[0,1)
C[0,2]
D[0,2)
9 ;<=>6 !"#$%&
é 9 "2024·ċú·õö÷ø% v*𝑀J,𝑥2 + 𝑦2 = 1-*I*𝑁J,𝐶:(𝑥−3)2 + 𝑦2 = 3-*I¤∠𝐶𝑀𝑁
. !" %
A
π
2
B
π
3
C
π
4
D
π
6
}\ 9-1 "2024·āĂ·õö÷ø% v'(𝑙:𝑥−𝑦+ 2 = 0?,𝑂:𝑥2 + 𝑦2 = 1I1'(𝑙-.ò¦*𝑃Ą,
𝑂.5( PAIPBI5*ºÎ AIBI¤∠𝐴𝑂𝐵. !" %
A
3π
4
B
2π
3
C
π
2
D
π
6
}\ 9-2 "23-24 Ė·óôüĥ·Ĩĩ%zP8'>;<T𝑥𝑂𝑦~I v 𝑂(0,0),𝐴15
4 ,0 IW(𝐶-ò
*𝑀ÕÖ|𝑂𝑀| = 4|𝐴𝑀|I*𝑃z'(𝑦=
2(𝑥−1)-IÒW(𝐶-ŇňzÛ*+*𝑃.xy2IćĈ*𝑃
.ʼn;<𝑡.#$J" %
A1 < 𝑡< 3
B1 < 𝑡< 4
C2 < 𝑡< 3
D2 < 𝑡< 4
}\ 9-3 "23-24 á-·ċúŊċ·Ĩĩ% v*𝑃z'(𝑙:3𝑥+ 4𝑦+ 3 = 0-I1𝑃Ą,𝑀:𝑥2 + 𝑦2
−6𝑥−4𝑦+ 9 = 0.Û£5(I5*𝐴,𝐵I¤∠𝐴𝑃𝐵. !" %
A30∘
B45∘
C60∘
D90∘
10 46 !"#$%&
é 10 "2024·ėĘŋŌ·õ%zP8'>;<T𝑥𝑂𝑦~I v𝐴(−3,0),𝐵(1,0),𝑃,𝐶:(𝑥−3)2 + (𝑦−3)2 = 1
-|*I¤|𝑃𝐴|2 + |𝑃𝐵|2. !" %
A34
B40
C44
D48
}\ 10-1 "23-24 âĖ·Ħħ·îïðñ% v,𝐶:𝑥2 + 𝑦2 = 4-Û*𝐴(𝑥1,𝑦1),𝐵(𝑥2,𝑦2)ÕÖ𝑥1𝑥2 + 𝑦1𝑦2
= 0I¤|𝑥1 +
3𝑦1 + 6| + |𝑥2 +
3𝑦2 + 6|. !" %
A3 2−2
B6−2 2
C6 2−4
D12−4 2
}\ 10-2 "2024·Ĝē¯·õö÷ø% v𝑃'(𝑙:𝑥+ 𝑦= 0-*I1*𝑃Ą,𝑀:(𝑥−1)2 + (𝑦−1)2 = 1
.5(𝑃𝐴"𝐴*5*%I𝐵,𝑁:(𝑥−3)2 + (𝑦−3)2 = 4-|* ¤|𝑃𝐴| + |𝑃𝐵|. !J" %
A 31−2
B3 2−1
C 31
D2 7−2
}\ 10-3 "23-24 â-·ġĢō·îïðñ% v,𝐶1:(𝑥−2)2 + (𝑦−3)2 = 1I,𝐶2:(𝑥−3)2 + (𝑦−4)2
= 9IMIN ºÎJ,𝐶1,𝐶2-.|*IP x ĺ-.|*I¤|𝑃𝑀| + |𝑃𝑁|. !" %
A5 2−2
B 17−1
C6 + 2 2
D5 2−4
=Ń
1"2024·ċú·õö÷ø% vĆ
𝑎,𝑏ÕÖ𝑎2 + 𝑏2 = 𝑎−𝑏I¤|𝑎+ 𝑏−3|. !" %
A 2
B2
C3 2
2
D4
2"2024·ĜēŎŏŐ·âõ%L'(𝑦= 𝑥-.*𝑃ij,(𝑥−4)2 + 𝑦2 = 4Ĵ5(I5*𝑄I¤|𝑃𝑄|.
!" %
A 2
B2
C 6
D2 2
3"2024·āĂ·õö÷ø%'(𝑦= 𝑘𝑥+ 2ı,𝑥2 + 𝑦2−6𝑥−7 = 0©.34. !" %
A 2
B 3
C2 2
D2 3
4"2024·ėĘęô·âõ%,(𝑥−1)2 + (𝑦+ 1)2 = 4-.*+'(3𝑥+ 4𝑦−14 = 0.xy. !" %
A3
B4
C5
D9
5"2024·ùúí~·áõ% v ⊙𝑀:𝑥2 + 𝑦2−2𝑥−2𝑦−2 = 0I'(𝑙:2𝑥+ 𝑦+ 2 = 0,𝑃 l -.|*IAIB
⊙𝑀-ò¦ÑĦY.Û*I¤cos∠𝐴𝑃𝐵. !" %
A−2 5
5
B−
4
5
C−5
5
D−
3
5
6"2024·Īő·õö÷ø% v* M J'(𝑙1:𝑎𝑥+ 𝑦−2𝑎= 0¡𝑙2:𝑥−𝑎𝑦+ 2 = 0"𝑎∈𝑅%.Ğ*I𝐴
(−1,0)I𝐵(𝑚,0)Ič* M ÕÖ|𝑀𝐴| =
1
2|𝑀𝐵|Į¯ÄIþ𝐶(2,2)I¤2|𝑀𝐴| + |𝑀𝐶|. !" %
A 6
B2 6
C 10
D2 10
7"23-24 á-·ĉĊċ·Ĩĩ% v'(𝑦= 𝑘𝑥+ 2(𝑘∈𝑅)Ğ,𝑂:𝑥2 + 𝑦2 = 9M𝑃(𝑥1,𝑦1),𝑄(𝑥2,𝑦2)Û*I¤
|3𝑥1 + 4𝑦1 + 16| + |3𝑥2 + 4𝑦2 + 16|. !" %
A9
B16
C27
D30
8"2024·ùúúĪ·õ% v,𝑂.[
𝑥2 + 𝑦2 = 1I*𝐴(2,0)I𝐵(0,2)I𝑃J(ï𝐴𝐵-.|*I1𝑃
Ą,𝑂.5(I5*ºÎ𝐶I𝐷I
@ĿĖĜrŒp
①Ĝŀ0𝑃𝐶𝑂𝐷.89. ! 1²②Ĝŀ0𝑃𝐶𝑂𝐷
.89. ! 3²③𝑃𝐶⋅𝑃𝐷. !−1²④𝑃𝐶⋅𝑃𝐷. !
3
2Ĕ~´@àœŒp.Ŕ
" %
A①③④
B①②④
C②③④
D①④
á=k
9"2024·ĪőŕĪ·õö÷ø% v,𝐶:𝑥2 + 𝑦2−4𝑥−5 = 0I*𝑃(𝑎,𝑏)J,𝐶-.*I¤ĖÇŒpàœ.
J" %
A,𝐶M'(𝑥−3𝑦−2 = 0mŖ
B v𝐴(1,−2)I𝐵(5,0)I¤|𝑃𝐴|2 + |𝑃𝐵|2. !32−12 2
C2𝑎+ 𝑏. !2−3 5
D
𝑎+2𝑏+9
𝑎+3 . !
19
4
10"2024·ŗŘřŀ·õ% v𝐴(𝑥1,𝑦1),𝐵(𝑥2,𝑦2)J,𝑂:𝑥2 + 𝑦2 = 4-.Û*I¤ĖÇX~àœ.J
" %
Aþ*𝑂+'(𝐴𝐵.xy 2I¤|𝐴𝐵| = 2 2
Bþ|𝐴𝐵| = 2 3I¤∠𝐴𝑂𝐵=
π
3
Cþ∠𝐴𝑂𝐵=
π
2I¤|𝑥1 + 𝑦1−1| + |𝑥2 + 𝑦2−1|. ! 6
D𝑥1𝑥2 + 𝑦1𝑦2. !−4
11"2024·ġĢŚĘ·õ% v,𝐶:(𝑥−2)2 + (𝑦−1)2 = 9I'(𝑙:𝑘𝑥−𝑦+ 1 = 0?𝐶ĞM𝐴,𝐵Û*I*𝑀
3𝐴𝐵.~*I𝑃(0,3)I¤" %
A3|𝐴𝐵|@ !2 5
B|𝑂𝑀|@ ! 2−1
C △𝑂𝐶𝑀89. ! 5+1
2
D𝑃𝑂⋅𝑃𝑀. ! 9
â=śŜ
12"2024·Ĝēŝİ·âõ%|'( l
𝑚𝑥+ 𝑦−2𝑚−1 = 0ı, C
𝑥2 + 𝑦2 +2𝑥−25 = 0©34. !
13"2024·ĜēŞĥ·õö÷ø%'(𝑙:𝑎𝑥+ 𝑦−2𝑎−2 = 0 (𝑎∈R)I?,𝐶:𝑥2 + 𝑦2−4𝑥−4𝑦= 1RĞM𝐴=𝐵
Û*I*𝑃'(𝑚:2𝑥+ 𝑦+ 6 = 0-|*I¤𝑃𝐴⋅𝑃𝐵. !J .
14"23-24 á-·ëş·Ĩĩ% v𝐴(𝑥1,𝑦1)=𝐵(𝑥2,𝑦2)ÕÖ
𝑥2
1 + 𝑦2
1 = 1I𝑥2
2 + 𝑦2
2 = 1I𝑥1𝑥2 + 𝑦1𝑦2 = −
1
2I¤¥
\|3𝑥1−4𝑦1| + |3𝑥2−4𝑦2|.Ď!#$J .
Ĝ=oŠ
15"23-24 áĖ·āĂ·šŢðñ% v*𝑃z,(𝑥−4)2 + (𝑦−5)2 = 16-I*𝐴(4,0)I𝐵(0,2)u*𝑃+'
(𝐴𝐵xy. !²
16"23-24 á-·āĂ·Ĩ~% v,𝑀1𝐴(−1,5)I𝐵(5,5)I𝐶(6,−2)â*.
(1)u,𝑀.<ţ[²
(2)þ*𝑃(𝑥,𝑦)z,𝑀-{|Iu2𝑥+ 𝑦. !.
17"2024 â·āĂ·ðñ% v𝑀(𝑚，𝑛),𝐶
𝑥2 + 𝑦2−4𝑥−14𝑦+ 45 = 0-ò¦*.
(1)u𝑚+ 2𝑛. !²
(2)u
𝑛−3
𝑚+2. !¡ !²
(3)u𝑚2 + 𝑛2. !¡ !.
18"23-24 á-·īĘŤť·Ĩĩ% v,𝐶:(𝑥−1)2 + (𝑦+ 1)2 = 4.
(1)1*𝑃(3,2)Ą𝐶.5(𝑙Iu𝑙.[²
(2)þ*𝑄'(𝑙′:3𝑥−4𝑦+ 13 = 0-.|*I1𝑄Ą,𝐶.5(IŦ5*𝑀Iŧ|𝑄𝑀|Ď !ÞIu∠𝐶𝑄𝑀
..
19"2024 â·āĂ·ðñ% v*𝑃(𝑥,𝑦)J,(𝑥+ 2)2 + 𝑦2 = 1-ò¦*
(1)u P *+'(3𝑥+ 4𝑦+ 12 = 0.xy. !¡ !
(2)u𝑥−2𝑦. !¡ !
(3)u
𝑦−2
𝑥−1. !¡ !.
