新高考數(shù)學(xué)二輪復(fù)習(xí)考點(diǎn)講義第二十五講直線方程及圓的方程（原卷版）

第二十五講：直線方程、圓的方程【考點(diǎn)梳理】直線的方程傾斜角、斜率，五種直線方程（點(diǎn)斜式、斜截式、兩點(diǎn)式、截距式、一般式）兩直線關(guān)系平行、垂直圓的方程（1）圓的標(biāo)準(zhǔn)方程：SKIPIF1<0，圓心坐標(biāo)為（a，b），半徑為SKIPIF1<0（2）圓的一般方程：SKIPIF1<0，圓心坐標(biāo)為SKIPIF1<0，半徑SKIPIF1<0直線與圓的位置關(guān)系幾何法、代數(shù)法（相離、相切、相交）兩圓的位置關(guān)系設(shè)兩個(gè)圓的半徑分別為SKIPIF1<0，SKIPIF1<0，圓心距為SKIPIF1<0，則兩圓的位置關(guān)系可用下表來表示：位置關(guān)系相離外切相交內(nèi)切內(nèi)含幾何特征SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0代數(shù)特征無實(shí)數(shù)解一組實(shí)數(shù)解兩組實(shí)數(shù)解一組實(shí)數(shù)解無實(shí)數(shù)解公切線條數(shù)43210【典型題型講解】考點(diǎn)一：直線的方程【典例例題】例1.若一次函數(shù)SKIPIF1<0所表示直線的傾斜角為SKIPIF1<0，則SKIPIF1<0的值為（

）．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例2.下列四個(gè)命題中真命題有_________個(gè)．①經(jīng)過定點(diǎn)SKIPIF1<0的直線都可以用方程SKIPIF1<0表示；②經(jīng)過任意兩點(diǎn)SKIPIF1<0的直線都可以用方程SKIPIF1<0表示；③不經(jīng)過原點(diǎn)的直線都可以用方程SKIPIF1<0表示；④經(jīng)過定點(diǎn)SKIPIF1<0的直線都可以用方程SKIPIF1<0表示．例3.已知SKIPIF1<0，SKIPIF1<0，則滿足SKIPIF1<0的SKIPIF1<0的值是（

）A．SKIPIF1<0 B．0 C．SKIPIF1<0或0 D．SKIPIF1<0或0例4.直線SKIPIF1<0和直線SKIPIF1<0垂直，則實(shí)數(shù)SKIPIF1<0__________．【方法技巧與總結(jié)】熟記直線方程的公式【變式訓(xùn)練】1.若圖中的直線l1，l2，l3的斜率分別為k1，k2，k3，則（）A．k1＜k2＜k3 B．k3＜k1＜k2C．k3＜k2＜k1 D．k1＜k3＜k22.已知集合SKIPIF1<0，集合SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0且SKIPIF1<0C．SKIPIF1<0且SKIPIF1<0 D．SKIPIF1<0且SKIPIF1<0且SKIPIF1<03.已知直線SKIPIF1<0恒過定點(diǎn)A，點(diǎn)A在直線SKIPIF1<0上，其中m、n均為正數(shù)，則SKIPIF1<0的最小值為（

）A．4 B．SKIPIF1<0 C．8 D．SKIPIF1<04.“SKIPIF1<0”是“直線SKIPIF1<0與直線SKIPIF1<0垂直”的（

）A．充分必要條件 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5.已知直線SKIPIF1<0：SKIPIF1<0．（1）求SKIPIF1<0經(jīng)過的定點(diǎn)坐標(biāo)SKIPIF1<0；（2）若直線SKIPIF1<0交SKIPIF1<0軸負(fù)半軸于點(diǎn)SKIPIF1<0，交SKIPIF1<0軸正半軸于點(diǎn)SKIPIF1<0．①SKIPIF1<0的面積為SKIPIF1<0，求SKIPIF1<0的最小值和此時(shí)直線SKIPIF1<0的方程；②當(dāng)SKIPIF1<0取最小值時(shí)，求直線SKIPIF1<0的方程．考點(diǎn)二：圓的方程【典例例題】例1．（2022·廣東·金山中學(xué)高三期末）“SKIPIF1<0”是“點(diǎn)SKIPIF1<0在圓SKIPIF1<0外”的（

）A．充分不必要條件 B．必要不充分條件C．充分必要條件 D．既不充分也不必要條件例2．（2022·廣東清遠(yuǎn)·高三期末）直線SKIPIF1<0被圓SKIPIF1<0截得的最短弦長為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例3．（2022·廣東·金山中學(xué)高三期末）（多選）已知點(diǎn)SKIPIF1<0，若過點(diǎn)SKIPIF1<0的直線SKIPIF1<0交圓SKIPIF1<0：SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0是圓SKIPIF1<0上一動點(diǎn)，則（

）A．SKIPIF1<0的最小值為SKIPIF1<0 B．SKIPIF1<0到SKIPIF1<0的距離的最大值為SKIPIF1<0C．SKIPIF1<0的最小值為SKIPIF1<0 D．SKIPIF1<0的最大值為SKIPIF1<0【方法技巧與總結(jié)】關(guān)于圓的切線的幾個(gè)重要結(jié)論（1）過圓SKIPIF1<0上一點(diǎn)SKIPIF1<0的圓的切線方程為SKIPIF1<0．（2）過圓SKIPIF1<0上一點(diǎn)SKIPIF1<0的圓的切線方程為SKIPIF1<0（3）過圓SKIPIF1<0上一點(diǎn)SKIPIF1<0的圓的切線方程SKIPIF1<0（4）求過圓SKIPIF1<0外一點(diǎn)SKIPIF1<0的圓的切線方程時(shí)，應(yīng)注意理解：①所求切線一定有兩條；②設(shè)直線方程之前，應(yīng)對所求直線的斜率是否存在加以討論．設(shè)切線方程為SKIPIF1<0，利用圓心到切線的距離等于半徑，列出關(guān)于SKIPIF1<0的方程，求出SKIPIF1<0值．若求出的SKIPIF1<0值有兩個(gè)，則說明斜率不存在的情形不符合題意；若求出的SKIPIF1<0值只有一個(gè)，則說明斜率不存在的情形符合題意．【變式訓(xùn)練】1．（2022·廣東廣州·二模）已知拋物線SKIPIF1<0，圓SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0交于A、B兩點(diǎn)，與SKIPIF1<0交于M、N兩點(diǎn)，若SKIPIF1<0，則SKIPIF1<0(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·廣東湛江·二模）已知直線SKIPIF1<0與圓SKIPIF1<0相交于A，B兩點(diǎn)，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·廣東梅州·二模）已知直線SKIPIF1<0與圓SKIPIF1<0交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，若SKIPIF1<0為等邊三角形，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022·廣東肇慶·二模）在SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，點(diǎn)D是線段AB上的動點(diǎn)﹐以D為圓心、AD長為半徑的圓與線段BC有公共點(diǎn)，則半徑AD的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．SKIPIF1<05．（2022·廣東·珠海市第三中學(xué)二模）已知圓SKIPIF1<0與拋物線SKIPIF1<0的準(zhǔn)線相切，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2022·廣東韶關(guān)·二模）已知直線SKIPIF1<0與圓SKIPIF1<0交于A、B兩點(diǎn)，若SKIPIF1<0則a=（

）A．5 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．（2022·廣東茂名·二模）（多選）已知a>0，圓C：SKIPIF1<0，則（

）A．存在3個(gè)不同的a，使得圓C與x軸或y軸相切B．存在2個(gè)不同的a，使得圓C在x軸和y軸上截得的線段相等C．存在2個(gè)不同的a，使得圓C過坐標(biāo)原點(diǎn)D．存在唯一的a，使得圓C的面積被直線SKIPIF1<0平分8．（2022·廣東·普寧市華僑中學(xué)二模）（多選）下列說法錯(cuò)誤的是（

）A．“SKIPIF1<0”是“直線SKIPIF1<0與直線SKIPIF1<0互相垂直”的充分必要條件B．直線SKIPIF1<0的傾斜角SKIPIF1<0的取值范圍是SKIPIF1<0C．若圓SKIPIF1<0與圓SKIPIF1<0有且只有一個(gè)公共點(diǎn)，則SKIPIF1<0D．若直線SKIPIF1<0與曲線SKIPIF1<0有公共點(diǎn)，則實(shí)數(shù)b的取值范圍是SKIPIF1<09．（2022·廣東深圳·二模）（多選）P是直線SKIPIF1<0上的一個(gè)動點(diǎn)，過點(diǎn)P作圓SKIPIF1<0的兩條切線，A，B為切點(diǎn)，則（

）A．弦長SKIPIF1<0的最小值為SKIPIF1<0 B．存在點(diǎn)P，使得SKIPIF1<0C．直線SKIPIF1<0經(jīng)過一個(gè)定點(diǎn) D．線段SKIPIF1<0的中點(diǎn)在一個(gè)定圓上10.（2022·廣東·二模）若直線SKIPIF1<0和直線SKIPIF1<0將圓SKIPIF1<0的周長四等分，則SKIPIF1<0__________．【鞏固練習(xí)】一、單選題1．已知P是半圓C：SKIPIF1<0上的點(diǎn)，Q是直線SKIPIF1<0上的一點(diǎn)，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知圓O：SKIPIF1<0，已知直線l：SKIPIF1<0與圓O的交點(diǎn)分別M，N，當(dāng)直線l被圓O截得的弦長最小時(shí)，SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．已知圓SKIPIF1<0截直線SKIPIF1<0所得的弦長為SKIPIF1<0，則圓C與圓SKIPIF1<0的位置關(guān)系是（

）A．相離 B．外切 C．相交 D．內(nèi)切4．設(shè)SKIPIF1<0，O為坐標(biāo)原點(diǎn)，點(diǎn)P滿足SKIPIF1<0，若直線SKIPIF1<0上存在點(diǎn)Q使得SKIPIF1<0，則實(shí)數(shù)k的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．點(diǎn)M為直線SKIPIF1<0上一點(diǎn)，過點(diǎn)M作圓O：SKIPIF1<0的切線MP，MQ，切點(diǎn)分別為P，Q，當(dāng)四邊形MPOQ的面積最小時(shí)，直線PQ的方程為（

）A．x＋y－2＝0 B．SKIPIF1<0C．x＋y－1＝0 D．x＋y＋1＝0二、多選題6．已知圓SKIPIF1<0的一般方程為SKIPIF1<0，則下列說法正確的是（

）A．圓SKIPIF1<0的圓心為SKIPIF1<0 B．圓SKIPIF1<0的半徑為5C．圓SKIPIF1<0被SKIPIF1<0軸截得的弦長為6 D．圓SKIPIF1<0被SKIPIF1<0軸截得的弦長為67.已知圓SKIPIF1<0被SKIPIF1<0軸分成兩部分的弧長之比為SKIPIF1<0，且被SKIPIF1<0軸截得的弦長為4，當(dāng)圓心SKIPIF1<0到直線SKIPIF1<0的距離最小時(shí)，圓SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．已知點(diǎn)SKIPIF1<0是圓SKIPIF1<0上的任意一點(diǎn)，直線SKIPIF1<0，則下列結(jié)論正確的是（

）A．直線SKIPIF1<0與圓SKIPIF1<0的位置關(guān)系只有相交和相切兩種B．圓SKIP