专题8.2两条直线的位置关系九大题型（原卷版）_11.4-2_2026-11-04

      
      
专题8.2 两条直线的位置关系【九大题型】

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 6  .56 !78.9: ....................................................................................................................5
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² 152024·³´µ·‹¶7() !𝑙1:2𝑥+ 𝑚𝑦−1 = 0，𝑙2:(𝑚+ 1)𝑥+ 3𝑦+ 1 = 0X£“𝑚= 2”–“𝑙1//
𝑙2”"5    7
A~·Cr¸>¹
B~¸>r·C¹
C~·>¹
D~ºr·C»r¸>¹
¼V 1-152024·½¾¾¿·À¶7().𝑀(𝑚,−1)X𝑁(4,𝑚)X† !𝑀𝑁% !2𝑥−𝑦+ 3 = 0& X£𝑚=
5    7
A~−6
B~
7
3
C~
2
3
D~9
¼V 1-252024·³´Á·¶ÃÄÅ7“𝑎= 0”–“ !𝑙1:𝑥+ 2𝑎𝑦−2024 = 0% !𝑙2
:(𝑎−1)𝑥+ 𝑎𝑦+ 2024 = 0#$”"5    7
A~·Cr¸>¹
B~¸>r·C¹
C~·>¹
D~ºr·C»r¸>¹
¼V 1-352024·³´·‹¶7() !𝐴𝑥+ 𝐵𝑦+ 𝐶= 0% !𝑦= 2𝑥−3& X£5    7
A~𝐴= −2𝐵≠0
B~𝐴= 2𝐵≠0
C~𝐵= −2𝐴≠0
D~𝐵= 2𝐴≠0
 2  '%() !#$*& " !+,
² 252024·ÆÇ·À¶7() !𝑙% !𝑥−𝑦= 0#$X†¦𝑦¬T"0–−2X£ !𝑙"+,–
5    7~
A~𝑥−𝑦+ 2 = 0
B~𝑥−2𝑦+ 4 = 0
C~𝑥−𝑦−2 = 0
D~𝑥+ 2𝑦−4 = 0
¼V 2-152024·ÈÇÉ[·¶ÃÄÅ7®.𝑃(−1,2)†% !𝑥+ 2𝑦+ 3 = 0& " !+,–5    7
A~𝑥−2𝑦+ 5 = 0
B~𝑥+ 2𝑦−3 = 0
C~2𝑥−𝑦+ 4 = 0
D~2𝑥+ 𝑦= 0
      
      
¼V 2-252024·ÊË·¶ÃÄÅ7 △𝐴𝐵𝐶šX𝐴(3,2)X𝐵(1,1)X𝐶(2,3)X£𝐴𝐵ÌT"ͦ" !+,
–5    7
A~2𝑥+ 𝑦−7 = 0
B~2𝑥−𝑦−1 = 0
C~𝑥+ 2𝑦−8 = 0
D~𝑥−2𝑦+ 4 = 0
¼V 2-3523-24 ÀT·ÈÇÎÏ·ÐÑ7®. −2，0 %𝑦= 𝑥#$" !+,–5    7
A~𝑥−𝑦−2 = 0
B~𝑥+ 𝑦+ 2 = 0
C~𝑥−𝑦+ 2 = 0
D~𝑥+ 𝑦−2 = 0
 3   !"-./
² 352024·[´[Ò·À¶7Ó !𝑦= −2𝑥+ 4% !𝑦= 𝑘𝑥"-.¦ !𝑦= 𝑥+ 2TX£œ
𝑘=
5    7
A~4
B~2
C~
1
2
D~
1
4
¼V 3-1523-24 ÀT·‡Ô˜Õ·ÐÑ7 !2𝑥−𝑦+ 6 = 0% !𝑥+ 𝑦= 3"-.OP–5   7
A~(3，0)
B~(−1,4)
C~(−3,6)
D~(4,−1)
¼V 3-2523-24 ÀT·Ö×ØÆ·ÐÑ7Ù® !2𝑥−3𝑦+ 10 = 0Ú3𝑥+ 4𝑦−2 = 0"-.X†&
 8 !2𝑥−𝑦−1 = 0" !+,…5    7
A~𝑥−2𝑦−6 = 0
B~𝑥+ 2𝑦−2 = 0
C~2𝑥−𝑦−3 = 0
D~2𝑥+ 𝑦−2 = 0
¼V 3-3523-24 ÀÛ·T[·Ðš7 !𝑙1:7𝑥+ 2𝑦+ 1 = 0,𝑙2:𝑚𝑥+ 𝑦= 0,𝑙3:𝑥+ 𝑚𝑦−1 = 0XӋ 
!ÜNÝދßoX£œ
𝑚Šà4"¢
…5    7
A~3
B~4
C~5
D~6
 4  01/
² 452024·áâ·¶ÃÄÅ7#$ !𝑙1:2𝑥+ 𝑦−5 = 0%𝑙2:𝑥−𝑏𝑦+ 5 = 0©U"01…5   7
A~ 5
B~2 5
C~3 5
D~5 5
¼V 4-152024·[´[Ò·¶ÃÄÅ7‰𝐴(0,
1
8)XÓã
𝑦= 𝑎𝑥2(𝑎> 0)äåT’“€.𝑃(𝑥0,𝑦0)æ›
|𝑃𝐴| = 𝑦0 +
1
8X£𝑎= 5    7
A~
1
4
B~
1
2
C~2
D~4
¼V 4-252024·³´·¶ÃÄÅ7()+,−𝑥2 +2𝑎𝑥+ 2 2𝑏= 2¦œ
çè2LX£𝑎2 + 𝑏2"
      
      
3é4…5    7
A~
1
2
B~
1
4
C~ 2
2
D~ 2
4
¼V 4-352024·Îê´_·€¶7()œ
𝑎> 0,𝑏< 0X£
3𝑏−𝑎
𝑎2+𝑏2"à4çè–5    7
A~[−2,−1)
B~(−2,−1)
C~(−2.−1]
D~[−2,−1]
 5  %012"34/
² 552024·ÊË·À¶7 !𝑙"+,…(𝜆+ 2)𝑥+ (𝜆−1)𝑦−3𝜆= 0(𝜆∈𝑅)Xƒ‘.𝑂W !𝑙"013s
wX𝜆"4…5    7
A~−1
B~−5
C~1
D~5
¼V 5-1523-24 ÀT·¿ë·ì—ív7Ó PXQ C… ! 3xî4yï12ð0 % 6xî8yî5ð0 T’“
€.X£|PQ|"3é4…(ññ)
A~
9
5 
B~
18
5 
C~
29
10 
D~
29
5 
¼V 5-2523-24 ‹T·‡Ô·ì—ív7¦#S ßOP;šXòˆ𝐴= {(𝑥,𝑦)|𝑘𝑥−𝑦+ 𝑘= 0 }Xòˆ
𝐵= {(𝑥,𝑦)|𝑦= 𝑘𝑥−1 }X().𝑀∈𝐴X.𝑁∈𝐵X„𝑑óô!—𝑀𝑁˜q"3é4X£𝑑"3s4…5    7
A~2
B~ 3
C~1
D~ 2
¼V 5-3523-24 ÀT·õöηК7÷ø
ùúûüýþ®
“
oÿˆĀāXĂăCùĄą
Ć~”ąœTX2ćjĈ
/Škĉ…bĊ/ykLċXČ
 (𝑥−𝑎)2 + (𝑦−𝑏)2Škĉ…#ST.
𝑀(𝑥,𝑦)%.𝑁(𝑎,𝑏)"01~ÿˆTčĎ.XŠď𝑦=
𝑥2−2𝑥+ 5 +
𝑥2−6𝑥+ 25"3é4…5    7
A~2 10
B~2 2
C~ 2 +
10
D~3 +
5
 6  .56 !78.9:
² 6523-24 ÀT·áâ·ÐÑ7.𝑃(1,2)¦ !𝑙TX !𝑙1%𝑙8.(0,1)9:X£€K¦ !𝑙1T"
.…5    7
A~ 1
2 , 3
2
B~ −1, 3
2
C~(−1,0)
D~51X07
¼V 6-1523-24 ÀT·Îê«Đ·Ðš7() !𝑥+ 2𝑦−3 = 0% !𝑎𝑥+ 4𝑦+ 𝑏= 08.𝐴(1,0)9
:X£œ
𝑏"4…5    7
A~2
B~6
C~−2
D~−6
¼V 6-2523-24 ÀT·^_[đ·Ðš7.𝑃(1,2)¦ !𝑙TX !𝑙1%𝑙8.(0,1)9:X£€K¦ 
      
      
!𝑙1T".…5  7
A~(
1
2,
3
2)
B~(−1,
3
2)
C~(−1,0)
D~(
1
2,0)
¼V 6-3523-24 €Û·Ēē°Ĕ·ÐÑ7% !3𝑥−4𝑦+ 5 = 08OP‘.9:" !+,…5    7
A~3𝑥+ 4𝑦−5 = 0
B~3𝑥+ 4𝑦+ 5 = 0
C~3𝑥−4𝑦+ 5 = 0
D~3𝑥−4𝑦−5 = 0
 7  .8 !9:
² 752024·ĕη¶ÃÄÅ7.(1,2)8 !𝑥+ 𝑦−2 = 0"9:.–5    7
A~(1,0)
B~(0,1)
C~(0,−1)
D~(2,1)
¼V 7-1523-24 ÀT·Ėė‹Ę·Ðš7()𝐴(−3,0),𝐵(0,3)X`.𝑃(−1,0)ęĚ"ě!Ù y ¬ĜęW 
!𝐴𝐵TXĝÙ® !𝐴𝐵ĜęW𝑃.X£ě!ÍÙ®"Ğ,…5    7
A~2 10
B~6
C~2 5
D~2 6
¼V 7-252024·½¾¾¿·€¶7ğĈĠġĢģ"ĠĤē`ĥ$ĦħĔĨþ
“ĩĪīÆĬĭĮXįİ
ıIJij-³”XĠšĴĵĶ€¢2ķ"
/——“ĸĥıIJ”/XĹĸĥ¦ĎĬĭĮ©ĺ`ÆĻÛļĽ
ĚľXĿW³ÌıIJĺŀŁWĥłXŃńŅņEŇňĞ,3¡ʼn¦#S ßOP;šX‰ĥłÍ¦"<=
…𝐴(−4,1)~Óĸĥ`ÆĻÛ".𝐵(−3,2)ĽĚľX³Ŋ!ͦ !+,…𝑥−𝑦+ 3 = 0X£“ĸĥıIJ”"3
¡ňĞ,…5    7
A~ 2
B~ 5
C~ 10
D~2 3
¼V 7-3523-24 ÀT·ĕÎŋŌ·Ðš7ČäX€ōě!`𝐴(1,0)ĚľXÙ !𝑥+ 𝑦+ 1 = 0Ĝęĺĝ
Ù®.𝐵(6,−5)X£ě!` A W B Ņ®"Ğ,…5    7
A~ 55
B~2 14
C~ 58
D~2 15
 8   !8 !"9:/
² 852024·T[Ŏ¿·À¶7‰ !𝑙1:𝑥−2𝑦−2 = 0%𝑙28 !𝑙:2𝑥−𝑦−4 = 09:X£ !𝑙2"+,–
5ññ7
      
      
A~11𝑥+ 2𝑦−22 = 0
B~11𝑥+ 𝑦+ 22 = 0
C~5𝑥+ 𝑦−11 = 0
D~10𝑥+ 𝑦−22 = 0
¼V 8-1523-24 ÀT·½¾¾¿·Ðš7‰ !𝑙1:3𝑥−2𝑦−6 = 0X !𝑙2:𝑥−𝑦−4 = 0X£𝑙18𝑙29:
" !+,…5    7
A~3𝑥+ 2𝑦−14 = 0
B~2𝑥−3𝑦−14 = 0
C~3𝑥+ 2𝑦−6 = 0
D~2𝑥−3𝑦−6 = 0
¼V 8-2523-24 ÀT·ŏ^Őő·ÐÑ7()ě!`.𝐴(−2,1)ęĚXÙ !2𝑥−𝑦+ 10 = 0ĜęX†Ĝ
ęě!ͦ !®.𝐵(−8,−3)X£Ĝęě!ͦ !"+,–5    7
A~𝑥−3𝑦−1 = 0
B~3𝑥−𝑦+ 21 = 0
C~𝑥+ 3𝑦+ 17 = 0
D~3𝑥+ 𝑦+ 15 = 0
¼V 8-3523-24 ÀT·ŏ^įŒ·ì—ív7Ó#$ !𝑙1
𝑥−2𝑦+ 𝑚= 0(𝑚> 0)%𝑙2
2𝑥+ 𝑛𝑦−6 = 0©U"01–2 5X£ !𝑙18 !𝑙29:" !+,…5    7
A~𝑥−2𝑦−13 = 0
B~𝑥−2𝑦+ 2 = 0
C~𝑥−2𝑦+ 4 = 0
D~𝑥−2𝑦−6 = 0
 9   !;+,
² 9523-24 ÀT·áâ·œĺŔŕ7® !𝑙1:𝑥−3𝑦+ 4 = 0Ú𝑙2:2𝑥+ 𝑦+ 5 = 0"-.ڑ." !
+,…5ññ7
A~3x-19y=0
B~19x-3y=0
C~19x+3y=0
D~3x+19y=0
¼V 9-1523-24 ÀT·‡Ô·ì—ív7Ù® !3𝑥+ 2𝑦+ 6 = 0Ú2𝑥+ 5𝑦−7 = 0"-.X†¦O
P¬T"0‚Ŗ" !+,…5    7
A~𝑥+ 𝑦+ 1 = 0
B~𝑥−𝑦+ 1 = 0
C~𝑥+ 𝑦+ 1 = 063𝑥+ 4𝑦= 0
D~𝑥−𝑦+ 1 = 06𝑥+ 𝑦+ 1 = 0
¼V 9-2523-24 ÀT·ŏ^ŗŘ·ì—ív7® !2023𝑥−2022𝑦−1 = 0Ú2022𝑥+ 2023𝑦+ 1 = 0
"-.†®‘." !+,…      
~
¼V 9-3523-24 ÀT·¿ëIJřƷК7#S ßOP;𝑥𝑂𝑦šX® !𝑙1:7𝑥−3𝑦+ 1 = 0%𝑙2
:𝑥+ 4𝑦−3 = 0"-.X†¦𝑦¬T0… 1 " !𝑙"+,…      
~5ŚÞ€V7
€*ś
      
      
1~52024·õöÎŜŝŞ·¶ÃÄÅ7() !𝑙1:𝑎𝑥+ 3𝑦−6 = 0X !𝑙2:2𝑥+ (𝑎−1)𝑦−4 = 0X£“𝑙1 ∥𝑙2”–
“𝑎= 36𝑎= −2”"5    7
A~·Cr¸>¹
B~¸>r·C¹
C~·>¹
D~ºr·C»r¸>¹
2~52024·õöÎÊË·À¶7#$ !𝑙1
𝑥+ 𝑦+ 1 = 0X𝑙2
𝑥+ 𝑦−1 = 0©U"01–5    7
A~1
B~ 2
C~2 2
D~2
3~52024·³´şĐ·¶ÃÄÅ7() !𝑙1:𝑥+ 𝑚𝑦+ 1 = 0% !𝑙2:𝑥+ (1−2𝑚)𝑦−3 = 0X£“𝑚∈{1,−2}”
–“𝑙1 ⊥𝑙2”"5    7
A~·Cr¸>¹
B~¸>r·C¹
C~·>¹
D~ºr·C»r¸>¹
4~52024·‡Ô·‹¶7ƒ.𝑃(−1,0)W ! l
(3𝜆+ 1)𝑥+ (𝜆+ 1)𝑦−(4𝜆+ 2) = 0"013swXœ
𝜆"4
…5    
7
A~−1
B~1
C~−2
D~2
5~52024·‡ÔŠšŢ·¶ÃÄÅ7‰ ! 𝑙:𝑥+ 𝑦−1 = 0X €ōě!`‘. 𝑂 Ěľţę! 𝑦= 𝑘𝑥(𝑥≥0) 
Ť ! 𝑙 ęĚX Ù 𝑙 Ĝęĺ% 𝑥 ¬-8. 𝑀X ŀťÙ 𝑥 ¬Ĝęĺ% 𝑦 ¬-8. 𝑁. Ó |𝑀𝑁| =
13
6 X 
£ 𝑘 "4…5    7
A~
3
2
B~
2
3
C~
1
2
D~
1
3
6~523-24 ÀT·‡ÔŦηì—ív7().𝐴(2,3),𝐵(4,1), !𝑥−2𝑦+ 4 = 0%𝑦¬‚-8.𝐶,£△𝐴𝐵𝐶š
𝐴𝐵ÌT"𝐶𝐸ͦ !"+,–5    7
A~𝑥+ 𝑦−2 = 0
B~𝑥+ 𝑦+ 2 = 0
C~𝑥−𝑦+ 2 = 0
D~𝑥−𝑦−2 = 0
7~52024·½¾¾¿·€¶7ğĈĠġĢģ"ĠĤē`ĥ$ĦħĔĨþ
“ĩĪīÆĬĭĮXįİıIJij-
³”XĠšĴĵĶ€¢2ķ"
/——“ĸĥıIJ”/XĹĸĥ¦ĎĬĭĮ©ĺ`ÆĻÛļĽĚľXĿ
W³ÌıIJĺŀŁWĥłXŃńŅņEŇňĞ,3¡ʼn¦#S ßOP;šX‰ĥłÍ¦"<=…𝐴
(−3,0)XÓĸĥ`ÆĻÛ".𝐵(−1,1)ĽĚľX³Ŋ!ͦ !+,…𝑥+ 𝑦= 1X£“ĸĥıIJ”"3¡ňĞ
,…5   7
A~ 5
B~3
C~ 13
D~5
      
      
8~52024·ŧĐŨũ·¶ÃÄÅ7 !𝑙1:𝑥+ (1 + 𝑎)𝑦= 1−𝑎(𝑎∈𝐑)X !𝑙2:𝑦= −
1
2𝑥XŪĚÛūŬ
①∃𝑎∈RXŇď𝑙1//𝑙2t    ②∃𝑎∈RXŇď𝑙1 ⊥𝑙2t
③∀𝑎∈RX𝑙1%𝑙2ŭ‚-t    ④∃𝑎∈RXŇď‘.W𝑙1"01…2.
™šŮů"–5        7
A~①②
B~②③
C~②④
D~①④
À*j
9~523-24 ÀT·ŰűĩŲ·ÐÑ7() !𝑙1:𝑥+ 2𝑦−2 = 0X !𝑙2:(𝑘−1)𝑥+ 𝑘𝑦+ 3 = 0X£5   7
A~ !𝑙2Šk%𝑥¬#$
B~ !𝑙2Šk%𝑦¬#$
C~ƒ𝑙1 ∥ 𝑙2wX𝑘= 2
D~ƒ𝑙1 ⊥𝑙2wX𝑘= −
1
3
10~523-24 ‹T·Îê·ì—ív7() !𝑙Ù®.(2,3)X†.𝐴(−3,2)X𝐵(5,−4)W !𝑙"01‚ŖX
£ !𝑙"+,ŠE…5    7
A~4𝑥−𝑦−5 = 0
B~4𝑥+ 𝑦−11 = 0
C~3𝑥+ 4𝑦−18 = 0
D~3𝑥−4𝑦+ 6 = 0
11~52024·ų´ŴĘ·¶ÃÄÅ7ğĈĠġĢģ"ĠĤē`ĥ$ĦħĔĨ
“ĩĪīÆĬĭĮXįİıIJ
ij-³”ĴŵĶ€¢2ķ"
/——“ĸĥıIJ”XĹļĸĥĎĬŶĭĮŷ©ĺ`ÆĻ"ļĽĚľXĿŸ
³ÌıIJXŀŹŁĥłXŃńŅEŇňĞ,3¡?¦#S ßOP;š2³ź mXnX™+,C…
2𝑥−𝑦= 0X𝑦= 0Xĸĥ"Ěľ.–.𝐴(3,1)XĥłÍ¦<=…𝐵(6,3)X£ÛūþNŻż"–5    7
A~ÓĸĥĿŸ³ź m ıIJXŀŹŁĥłX£ĸĥ¦³ÌıIJ"."OP…(1,2)
B~ĸĥĿŸ³ź n ıIJXŀŹŁĥł"3¡Ğ,– 5
C~ĸĥĿŸ³ź m ıIJXŀŸ³ź n ıIJX3口ĥł"3¡Ğ,– 85
D~ĸĥĿŸ³ź n ıIJXŀŸ³ź m ıIJX3口ĥł"3¡Ğ,–2 13
‹*mn
12~52024·ÆÇ·À¶7® !𝑥+ 𝑦+ 1 = 0Ú3𝑥−𝑦−3 = 0"-.XŽH߅45°" !+,…      
.
13~52024·½¾žÁ·¶ÃÄÅ7() !𝑙1:−2𝑥+ (𝑎−2)𝑦+ 3 = 0% !𝑙2:𝑏𝑥−𝑦−1 = 0,𝑎> 0,𝑏> 0XÓ
𝑙1 ⊥𝑙2X£
𝑎𝑏
𝑎+2𝑏"3s4…      .
14~52024·Ö×Þŭ·¶ÃÄÅ7() !𝑙Ù®.𝑃(0,1)X†ſ#$ !𝑙1: 3𝑥+ 𝑦+ 1 = 0Ú𝑙2: 3
𝑥+ 𝑦+ 5 = 0ď"!—˜…2 2X£ !𝑙"+,…      
~
Ö*Lƀ
      
      
15~523-24 ÀÛ·Ö×Ɓ¿·ħƂ7() !𝑙1:2𝑥+ (𝑎−1)𝑦+ 1 = 0X !𝑙2:(𝑎+ 4)𝑥+ 3𝑦+ 3 = 0~
(1)Ó𝑙1//𝑙2X'œ
𝑎"4t
(2)Ó𝑙1 ⊥𝑙2X'œ
𝑎"4~
16~52024·½¾¾¿·À¶7LƀÛū/.
(1)() !𝑙1:𝑎𝑥−3𝑦+ 4𝑏= 0% !𝑙2:2𝑥+ 𝑏𝑦−2𝑎= 0‚-X-.OP…(1,2)X'𝑎,𝑏"4t
(2)() !𝑙®.𝑃(2,3)X†.𝑀(3,1)W !𝑙"01…1X' !𝑙"+,.
17~523-24 ÀT·ĕÎƃú·Ðš7() !𝑙1:3𝑥−𝑦−1 = 0,𝑙2:𝑥+ 2𝑦−5 = 0.
(1)'® !"-.X†& 8 !3𝑥+ 4𝑦−5 = 0" !+,t
(2)().𝐴(−1,1),𝐵(0,2)XƄ.𝑃¦ !𝑙1ƅƄX'|𝑃𝐴| + |𝑃𝐵|"3é4.
18~524-25 ÀT·T[·ƆƇív7ČäX()𝐴(6,6 3)X𝐵(0,0)X𝐶(12,0)X !𝑙
(𝑘+
3)
𝑥−𝑦−2𝑘= 0~
(1)' !𝑙Ù®"K.OPt
      
      
(2)Ó𝑃(2,2 3)XĢƈƉƊ¦.𝑃Ƌěƌƍ̀ōě!XƎťƏ𝐵𝐶5Ĝę.…𝐾7*𝐴𝐶5Ĝę.…𝐼7Ĝę
ĺXěƐƑ¦𝑃.X'ƒęě!𝑃𝐾" !+,~
19~523-24 ÀÛ·T[·ì—ív7() △𝐴𝐵𝐶"‹¢Ɠ."OPC–.𝐴(5,1)、𝐵(4,3)%𝐶(0,−1)X 
!𝑚:(𝑘+ 2)𝑥+ (𝑘−1)𝑦+ 𝑘−1 = 0    (𝑘∈𝑅).
(1)'Ì AC ͦ !𝑙1"ŽHßÚÌ AC T"ͦ !𝑙2"+,;
(2)„𝑑….𝐴W !𝑚"01XƂ/:𝑑–Ɣƕ¦3s4?Óƕ¦X'Ě𝑑"3s4:Órƕ¦XþĘƏ;