第二章2.2.3兩條直線的位置關(guān)系A(chǔ)級(jí)必備知識(shí)基礎(chǔ)練12345678910111213141516171819201.[探究點(diǎn)三](多選題)已知直線l1的斜率為a,l1⊥l2,則l2的斜率可以為(

)21BD解析

當(dāng)a≠0時(shí),由k1k2=-1知,k2=-,當(dāng)a=0時(shí),l2的斜率不存在.1234567891011121314151617181920212.[探究點(diǎn)一]下列四組直線中,互相垂直的一組是(

)A.2x+y-1=0與2x-y-1=0B.2x+y-1=0與x-2y+1=0C.x+2y-1=0與x-y-1=0D.x+y=0與x+y-3=0B解析

對(duì)于A,2x+y-1=0與2x-y-1=0,有2×2+1×(-1)≠0,兩直線不垂直,不符合題意;對(duì)于B,2x+y-1=0與x-2y+1=0,有2×1+1×(-2)=0,兩直線垂直,符合題意;對(duì)于C,x+2y-1=0與x-y-1=0,有1×1+2×(-1)≠0,兩直線不垂直,不符合題意;對(duì)于D,x+y=0與x+y-3=0,兩直線平行,不符合題意.故選B.1234567891011121314151617181920213.[探究點(diǎn)三]已知點(diǎn)A(1,2),B(3,1),則線段AB的垂直平分線的方程是(

)A.4x+2y=5 B.4x-2y=5C.x+2y=5 D.x-2y=5B1234567891011121314151617181920214.[探究點(diǎn)一](多選題)下列說(shuō)法中,正確的是(

)A.直線2x+y+3=0在y軸上的截距是3B.直線x+y+1=0的傾斜角為135°C.A(1,4),B(2,7),C(-3,-8)三點(diǎn)共線D.直線3x+4y+1=0與4x+3y+2=0垂直BC解析

直線2x+y+3=0在y軸上的截距是-3,故A錯(cuò)誤;直線x+y+1=0的斜率為-1,傾斜角為135°,故B正確;1234567891011121314151617181920215.[探究點(diǎn)一](多選題)設(shè)平面內(nèi)四點(diǎn)P(-4,2),Q(6,-4),R(12,6),S(2,12),下面四個(gè)結(jié)論正確的是(

)A.PQ∥SR B.PQ⊥PS C.PS∥QS D.PR⊥QSABD∴PS與QS不平行.故ABD正確.1234567891011121314151617181920216.[探究點(diǎn)二]經(jīng)過(guò)兩條直線2x+3y+1=0和2x-3y+3=0的交點(diǎn),并且平行于直線y=x的直線的一般式方程為

.

3x-3y+4=01234567891011121314151617181920217.[探究點(diǎn)二、三]設(shè)直線l1:(a+1)x+3y+2-a=0,直線l2:2x+(a+2)y+1=0,若l1⊥l2,則實(shí)數(shù)a的值為

;若l1∥l2,則實(shí)數(shù)a的值為

.

-4解析

若l1⊥l2,則2(a+1)+3(a+2)=0,整理可得5a+8=0,求解關(guān)于實(shí)數(shù)a的方1234567891011121314151617181920218.[探究點(diǎn)三]已知△ABC的三個(gè)頂點(diǎn)A(1,1),B(4,0),C(3,2),求BC邊上的高所在的直線方程及高的長(zhǎng)度.1234567891011121314151617181920211234567891011121314151617181920219.[探究點(diǎn)一·北師大版教材例題]已知A(1,4),B(-2,-1),C(4,1)是△ABC的三個(gè)頂點(diǎn),求證:△ABC的三條中線交于一點(diǎn).證明

根據(jù)已知條件將A,B,C三點(diǎn)畫(huà)在平面直角坐標(biāo)系中,如圖.設(shè)點(diǎn)E,F,G分別為AB,BC,AC的中點(diǎn),則易求得三邊的中點(diǎn)坐標(biāo)分別為123456789101112131415161718192021中線CE所在直線上.所以△ABC的三條中線交于一點(diǎn).1234567891011121314151617181920B級(jí)關(guān)鍵能力提升練2110.已知直線l1:xsinα+y-1=0,直線l2:x-3ycosα+1=0.若l1⊥l2,則sin2α=(

)A解析

∵l1⊥l2,∴sin

α-3cos

α=0,即tan

α=3.12345678910111213141516171819202111.“m=2”是“直線2x+(m+1)y+4=0與直線3x-my-2=0垂直”的(

)A.充要條件

B.充分不必要條件C.必要不充分條件 D.既不充分也不必要條件B解析

直線2x+(m+1)y+4=0與直線3x-my-2=0垂直,則2×3+(m+1)×(-m)=0,解得m=2或m=-3,所以“m=2”是“直線2x+(m+1)y+4=0與直線3x-my-2=0垂直”的充分不必要條件.故選B.12345678910111213141516171819202112.已知直線ax+2y+6=0與直線x+(a-1)y+a2-1=0互相平行,則實(shí)數(shù)a的值為(

)A.-2 B.2或-1 C.2 D.-1D解析

直線ax+2y+6=0斜率必存在,故兩直線平行,則

,即a2-a-2=0,解得a=2或-1,當(dāng)a=2時(shí),兩直線重合,故a=-1.故選D.12345678910111213141516171819202113.直線2ax+y-2=0與直線x-(a2-3)y+2=0互相垂直,且兩直線交點(diǎn)位于第三象限,則實(shí)數(shù)a的值為(

)A.1 B.3 C.-1 D.-3C解析

由直線2ax+y-2=0與直線x-(a2-3)y+2=0互相垂直,可得2a-(a2-3)=0,解得a=-1或3.12345678910111213141516171819202114.若點(diǎn)P(a,b)與點(diǎn)Q(b-1,a+1)關(guān)于直線l對(duì)稱(chēng),則直線l的傾斜角α為

.

45°由題意知PQ⊥l,則kPQ·kl=-1,得kl=1,∴直線l的傾斜角為45°.12345678910111213141516171819202115.若三條直線2x-y+4=0,x-y+5=0和2mx-3y+12=0圍成直角三角形,則m=

.

解析

設(shè)l1:2x-y+4=0,l2:x-y+5=0,l3:2mx-3y+12=0,l1不垂直于l2,要使圍成的三角形為直角三角形,則l3⊥l1或l3⊥l2.12345678910111213141516171819202116.已知集合A={(x,y)|2x-(a+1)y-1=0},B={(x,y)|ax-y+1=0},且A∩B=?,則實(shí)數(shù)a的值為

.

1解析

∵集合A={(x,y)|2x-(a+1)y-1=0},B={(x,y)|ax-y+1=0},且A∩B=?,∴直線2x-(a+1)y-1=0與直線ax-y+1=0平行,即-2=-a(a+1),且2≠-a,解得a=1.12345678910111213141516171819202117.求經(jīng)過(guò)兩條直線2x-3y-3=0和x+y+2=0的交點(diǎn)且與直線3x+y-1=0垂直的直線方程.即所求的直線方程為5x-15y-18=0.12345678910111213141516171819202118.已知M(1,-1),N(2,2),P(3,0).(1)求點(diǎn)Q的坐標(biāo),滿足PQ⊥MN,PN∥MQ;(2)若點(diǎn)Q在x軸上,且∠NQP=∠NPQ,求直線MQ的傾斜角.聯(lián)立①②解得x=0,y=1,∴Q(0,1).123456789101112131415161718192021(2)設(shè)Q(x,0).∵∠NQP=∠NPQ,∴kNQ=-kNP.又M(1,-1),∴MQ⊥x軸,故直線MQ的傾斜角為90°.12345678910111213141516171819202119.

如圖所示,一個(gè)矩形花園里需要鋪設(shè)兩條筆直的小路,已知矩形花園長(zhǎng)|AD|=5m,寬|AB|=3m,其中一條小路定為AC,另一條小路過(guò)點(diǎn)D,如何在BC上找到一點(diǎn)M,使得AC與DM兩條小路互相垂直?解

如圖所示,以點(diǎn)B為原點(diǎn),分別以BC,BA所在直線為x軸、y軸建立平面直角坐標(biāo)系,單位:m.由|AD|=5

m,|AB|=3

m得C(5,0),D(5,3),A(0,3).設(shè)點(diǎn)M的坐標(biāo)為(x,0),∵AC⊥DM,故當(dāng)|BM|=3.2

m時(shí),兩條小路AC與DM互相垂直.1234567891011121314151617181920C級(jí)學(xué)科素養(yǎng)創(chuàng)新練2120.直線l的傾斜角為30°,點(diǎn)P(2,1)在直線l上,直線l繞點(diǎn)P(2,1)按逆時(shí)針?lè)较蛐D(zhuǎn)30°后到達(dá)直線l1的位置,此時(shí)直線l1與l2平行,且l2是線段AB的垂直平分線,其中A(1,m-1),B(m,2),則m=

.

12345678910111213141516171819202121.已知點(diǎn)A(4,-1)和點(diǎn)B(8,2)均在直線l:x-y-