人教A版高中數(shù)學(xué)(選擇性必修第一冊(cè))同步講義第20講 2.4.1圓的標(biāo)準(zhǔn)方程（原卷版）

第07講2.4.1圓的標(biāo)準(zhǔn)方程課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①理解圓的定義及確定圓的幾何要素。②理解與掌握平面直角坐標(biāo)系中圓的標(biāo)準(zhǔn)方程.。③會(huì)根據(jù)相關(guān)條件寫(xiě)出圓的標(biāo)準(zhǔn)方程及圓的圓心，半徑。通過(guò)本節(jié)課的學(xué)習(xí)，了解與掌握確定圓的位置，大小的幾何要素，能根據(jù)相關(guān)條件求出圓的標(biāo)準(zhǔn)方程，并能解決與圓有關(guān)的問(wèn)題.知識(shí)點(diǎn)01：圓的定義平面內(nèi)到定點(diǎn)的距離等于定長(zhǎng)的點(diǎn)的集合叫作圓，定點(diǎn)稱(chēng)為圓心，定長(zhǎng)稱(chēng)為圓的半徑.如圖，在平面直角坐標(biāo)系中，SKIPIF1<0的圓心SKIPIF1<0的坐標(biāo)為SKIPIF1<0,半徑為SKIPIF1<0,SKIPIF1<0為圓上任意一點(diǎn)，SKIPIF1<0可用集合表示為：SKIPIF1<0知識(shí)點(diǎn)02：圓的標(biāo)準(zhǔn)方程我們把方程SKIPIF1<0稱(chēng)為圓心為SKIPIF1<0半徑為SKIPIF1<0的圓的標(biāo)準(zhǔn)方程.【即學(xué)即練1】（2023春·重慶沙坪壩·高一重慶八中校考期末）在平面直角坐標(biāo)系SKIPIF1<0中，已知SKIPIF1<0、SKIPIF1<0兩點(diǎn)，若圓SKIPIF1<0以SKIPIF1<0為直徑，則圓SKIPIF1<0的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【答案】A【詳解】由題意可知，圓心SKIPIF1<0的橫坐標(biāo)為SKIPIF1<0，縱坐標(biāo)為SKIPIF1<0，即點(diǎn)SKIPIF1<0，圓SKIPIF1<0的半徑為SKIPIF1<0，因此，圓SKIPIF1<0的標(biāo)準(zhǔn)方程為SKIPIF1<0.故選：A.知識(shí)點(diǎn)03：點(diǎn)與圓的位置關(guān)系判斷點(diǎn)SKIPIF1<0與SKIPIF1<0：SKIPIF1<0位置關(guān)系的方法:（1）幾何法（優(yōu)先推薦）設(shè)SKIPIF1<0到圓心SKIPIF1<0的距離為SKIPIF1<0，則SKIPIF1<0①SKIPIF1<0SKIPIF1<0則點(diǎn)SKIPIF1<0在SKIPIF1<0外②SKIPIF1<0SKIPIF1<0則點(diǎn)SKIPIF1<0在SKIPIF1<0上③SKIPIF1<0SKIPIF1<0則點(diǎn)SKIPIF1<0在SKIPIF1<0內(nèi)（2）代數(shù)法將點(diǎn)SKIPIF1<0帶入SKIPIF1<0：SKIPIF1<0方程內(nèi)①點(diǎn)SKIPIF1<0在SKIPIF1<0外SKIPIF1<0SKIPIF1<0②點(diǎn)SKIPIF1<0在SKIPIF1<0上SKIPIF1<0SKIPIF1<0③點(diǎn)SKIPIF1<0在SKIPIF1<0內(nèi)SKIPIF1<0SKIPIF1<0【即學(xué)即練2】（2023·江蘇·高二假期作業(yè)）寫(xiě)出圓心為SKIPIF1<0,半徑為5的圓的標(biāo)準(zhǔn)方程,并判斷點(diǎn)SKIPIF1<0是否在這個(gè)圓上．若該點(diǎn)不在圓上,說(shuō)明該點(diǎn)在圓外還是在圓內(nèi)？【答案】答案見(jiàn)解析【詳解】圓心為SKIPIF1<0,半徑為5的圓的標(biāo)準(zhǔn)方程是SKIPIF1<0．把點(diǎn)SKIPIF1<0的坐標(biāo)代入方程SKIPIF1<0的左邊,得SKIPIF1<0,左右兩邊相等,點(diǎn)SKIPIF1<0的坐標(biāo)滿(mǎn)足圓的方程,所以點(diǎn)SKIPIF1<0在這個(gè)圓上．把點(diǎn)SKIPIF1<0的坐標(biāo)代入方程SKIPIF1<0的左邊,得SKIPIF1<0,左右兩邊不相等,點(diǎn)SKIPIF1<0的坐標(biāo)不滿(mǎn)足圓的方程,所以點(diǎn)SKIPIF1<0不在這個(gè)圓上．又因?yàn)辄c(diǎn)SKIPIF1<0到圓心A的距離SKIPIF1<0SKIPIF1<0．故點(diǎn)SKIPIF1<0在圓內(nèi)．知識(shí)點(diǎn)04：圓上的點(diǎn)到定點(diǎn)的最大、最小距離設(shè)SKIPIF1<0的方程SKIPIF1<0，圓心SKIPIF1<0，點(diǎn)SKIPIF1<0是SKIPIF1<0上的動(dòng)點(diǎn)，點(diǎn)SKIPIF1<0為平面內(nèi)一點(diǎn)；記SKIPIF1<0;①若點(diǎn)SKIPIF1<0在SKIPIF1<0外，則SKIPIF1<0;SKIPIF1<0②若點(diǎn)SKIPIF1<0在SKIPIF1<0上，則SKIPIF1<0;SKIPIF1<0③若點(diǎn)SKIPIF1<0在SKIPIF1<0內(nèi)，則SKIPIF1<0;SKIPIF1<0【即學(xué)即練3】（2021秋·高二課時(shí)練習(xí)）已知圓SKIPIF1<0，則圓上的點(diǎn)到點(diǎn)SKIPIF1<0距離的最大值為_(kāi)____.【答案】6【詳解】因?yàn)閳ASKIPIF1<0的方程為SKIPIF1<0，所以圓心坐標(biāo)為SKIPIF1<0，半徑SKIPIF1<0,又圓心SKIPIF1<0到點(diǎn)SKIPIF1<0的距離為SKIPIF1<0，所以圓上的點(diǎn)到點(diǎn)SKIPIF1<0的距離的最大值為SKIPIF1<0，故答案為：6題型01求圓的標(biāo)準(zhǔn)方程【典例1】（2023·高二課時(shí)練習(xí)）已知圓C：SKIPIF1<0，O為原點(diǎn)，則以SKIPIF1<0為直徑的圓方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·上海崇明·高二統(tǒng)考期末）已知兩點(diǎn)SKIPIF1<0、SKIPIF1<0，則以PQ為直徑的圓的方程是______.【變式1】（2023·江蘇·高二假期作業(yè)）圓心在SKIPIF1<0軸上，半徑為5，且過(guò)點(diǎn)SKIPIF1<0，則圓的標(biāo)準(zhǔn)方程為_(kāi)______．【變式2】（2023·陜西西安·校聯(lián)考模擬預(yù)測(cè)）過(guò)三點(diǎn)SKIPIF1<0、SKIPIF1<0、SKIPIF1<0的圓的圓心坐標(biāo)為_(kāi)__________.題型02由圓的方程求圓心或半徑【典例1】（2023·江蘇·高二假期作業(yè)）下列說(shuō)法錯(cuò)誤的是（

）A．圓SKIPIF1<0的圓心為SKIPIF1<0，半徑為5B．圓SKIPIF1<0的圓心為SKIPIF1<0，半徑為SKIPIF1<0C．圓SKIPIF1<0的圓心為SKIPIF1<0，半徑為SKIPIF1<0D．圓SKIPIF1<0的圓心為SKIPIF1<0，半徑為SKIPIF1<0【典例2】（2023·全國(guó)·高三專(zhuān)題練習(xí)）已知圓SKIPIF1<0關(guān)于直線(xiàn)SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）對(duì)稱(chēng)，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．9 C．4 D．8【變式1】（2023·寧夏銀川·六盤(pán)山高級(jí)中學(xué)校考三模）已知直線(xiàn)SKIPIF1<0經(jīng)過(guò)圓SKIPIF1<0的圓心，其中SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．7 B．8 C．9 D．12【變式2】（2023·全國(guó)·高三專(zhuān)題練習(xí)）已知直線(xiàn)SKIPIF1<0過(guò)圓SKIPIF1<0的圓心，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．2題型03點(diǎn)與圓的位置關(guān)系【典例1】（2023·全國(guó)·高三專(zhuān)題練習(xí)）已知兩直線(xiàn)SKIPIF1<0與SKIPIF1<0的交點(diǎn)在圓SKIPIF1<0的內(nèi)部，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）.A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023·江蘇·高二假期作業(yè)）點(diǎn)SKIPIF1<0與圓SKIPIF1<0的位置關(guān)系是（

）A．在圓外 B．在圓內(nèi) C．在圓上 D．不確定【變式1】（多選）（2023·全國(guó)·高二專(zhuān)題練習(xí)）（多選）點(diǎn)SKIPIF1<0在圓SKIPIF1<0的內(nèi)部，則SKIPIF1<0的取值不可能是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023·全國(guó)·高三專(zhuān)題練習(xí)）若點(diǎn)SKIPIF1<0在圓SKIPIF1<0內(nèi)，則實(shí)數(shù)SKIPIF1<0的取值范圍為_(kāi)___________.題型04與圓有關(guān)的最值問(wèn)題【典例1】（2023·廣東佛山·統(tǒng)考模擬預(yù)測(cè)）已知圓SKIPIF1<0：SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的兩條直線(xiàn)SKIPIF1<0，SKIPIF1<0互相垂直，圓心SKIPIF1<0到直線(xiàn)SKIPIF1<0，SKIPIF1<0的距離分別為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的最大值為（

）A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．4【典例2】（2023秋·四川巴中·高二統(tǒng)考期末）已知圓C過(guò)點(diǎn)SKIPIF1<0，當(dāng)圓SKIPIF1<0到原點(diǎn)SKIPIF1<0的距離最小時(shí)，圓SKIPIF1<0的標(biāo)準(zhǔn)方程為_(kāi)_____．【變式1】（2023春·廣西·高一校聯(lián)考階段練習(xí)）若復(fù)數(shù)SKIPIF1<0滿(mǎn)足SKIPIF1<0，則SKIPIF1<0的最大值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．7 D．SKIPIF1<0【變式2】（2023·甘肅酒泉·統(tǒng)考三模）點(diǎn)SKIPIF1<0在圓SKIPIF1<0上，點(diǎn)SKIPIF1<0，則SKIPIF1<0的最大值為（

）A．3 B．4 C．5 D．6題型05與圓有關(guān)的對(duì)稱(chēng)問(wèn)題【典例1】（2023秋·四川成都·高二統(tǒng)考期末）已知圓SKIPIF1<0和直線(xiàn)SKIPIF1<0.若圓SKIPIF1<0與圓SKIPIF1<0關(guān)于直線(xiàn)SKIPIF1<0對(duì)稱(chēng)，則圓SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·四川涼山·高二?？茧A段練習(xí)）若圓SKIPIF1<0和圓SKIPIF1<0關(guān)于直線(xiàn)SKIPIF1<0對(duì)稱(chēng)，則直線(xiàn)SKIPIF1<0的方程是___________【變式1】（2023秋·四川成都·高二統(tǒng)考期末）已知圓SKIPIF1<0和直線(xiàn)SKIPIF1<0.若圓SKIPIF1<0與圓SKIPIF1<0關(guān)于直線(xiàn)SKIPIF1<0對(duì)稱(chēng)，則圓SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023秋·云南昆明·高二統(tǒng)考期末）已知圓SKIPIF1<0的圓心坐標(biāo)為SKIPIF1<0，半徑為2，圓SKIPIF1<0與圓SKIPIF1<0關(guān)于SKIPIF1<0軸對(duì)稱(chēng)，則圓SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型06軌跡方程【典例1】（2023秋·高一單元測(cè)試）已知定點(diǎn)SKIPIF1<0，SKIPIF1<0是圓SKIPIF1<0上的一動(dòng)點(diǎn)，SKIPIF1<0是SKIPIF1<0的中點(diǎn)，則點(diǎn)SKIPIF1<0的軌跡方程是_______________.【典例2】（2023秋·浙江麗水·高二統(tǒng)考期末）在平面直角坐標(biāo)系中，已知點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0在圓SKIPIF1<0上運(yùn)動(dòng)，則線(xiàn)段SKIPIF1<0的中點(diǎn)SKIPIF1<0的軌跡方程是______．【典例3】（2023·全國(guó)·高三專(zhuān)題練習(xí)）在直角坐標(biāo)系SKIPIF1<0中，線(xiàn)段SKIPIF1<0，且兩個(gè)端點(diǎn)SKIPIF1<0、SKIPIF1<0分別在SKIPIF1<0軸和SKIPIF1<0軸上滑動(dòng)．求線(xiàn)段SKIPIF1<0的中點(diǎn)SKIPIF1<0的軌跡方程；【變式1】（2023春·福建福州·高二福建省福州延安中學(xué)?？茧A段練習(xí)）已知線(xiàn)段SKIPIF1<0的端點(diǎn)SKIPIF1<0的坐標(biāo)是SKIPIF1<0，端點(diǎn)SKIPIF1<0在圓SKIPIF1<0上運(yùn)動(dòng)，則線(xiàn)段SKIPIF1<0的中點(diǎn)SKIPIF1<0的軌跡方程是__________．【變式2】（2023春·甘肅金昌·高二永昌縣第一高級(jí)中學(xué)校考階段練習(xí)）已知圓心為SKIPIF1<0的圓經(jīng)過(guò)SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且圓心SKIPIF1<0在直線(xiàn)SKIPIF1<0上.(1)求圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)設(shè)SKIPIF1<0為圓SKIPIF1<0上的一個(gè)動(dòng)點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，求SKIPIF1<0的中點(diǎn)SKIPIF1<0的軌跡方程.A夯實(shí)基礎(chǔ)B能力提升C綜合素養(yǎng)A夯實(shí)基礎(chǔ)一、單選題1．（2023春·上海徐匯·高二上海中學(xué)?？计谥校┮阎粋€(gè)圓的方程滿(mǎn)足：圓心在點(diǎn)SKIPIF1<0，且過(guò)原點(diǎn)，則它的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2023·北京海淀·校考三模）若直線(xiàn)SKIPIF1<0是圓SKIPIF1<0的一條對(duì)稱(chēng)軸，則SKIPIF1<0（

）A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．SKIPIF1<03．（2023春·四川南充·高二校考階段練習(xí)）圓SKIPIF1<0的圓心、半徑是（）A．SKIPIF1<0，4 B．SKIPIF1<0，2 C．SKIPIF1<0，4 D．SKIPIF1<0，24．（2023春·新疆省直轄縣級(jí)單位·高二校考開(kāi)學(xué)考試）已知點(diǎn)（1，1）在圓（x﹣a）2+（y+a）2＝4的內(nèi)部，則實(shí)數(shù)a的取值范圍是（

）A．（﹣1，1） B．（0，1）C．（﹣∞，﹣1）∪（1，+∞） D．{1，﹣1}5．（2023秋·河北保定·高二統(tǒng)考期末）圓SKIPIF1<0關(guān)于直線(xiàn)SKIPIF1<0對(duì)稱(chēng)的圓的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．（2023春·安徽·高二校聯(lián)考開(kāi)學(xué)考試）在圓的方程的探究中，有四位同學(xué)分別給出了一個(gè)結(jié)論，甲：該圓的半徑為SKIPIF1<0；乙：該圓經(jīng)過(guò)點(diǎn)SKIPIF1<0；丙：該圓的圓心為SKIPIF1<0；?。涸搱A經(jīng)過(guò)點(diǎn)SKIPIF1<0．如果只有一位同學(xué)的結(jié)論是錯(cuò)誤的，那么這位同學(xué)是（

）A．甲 B．乙 C．丙 D．丁7．（2023春·甘肅蘭州·高二統(tǒng)考期中）已知圓SKIPIF1<0，則過(guò)點(diǎn)SKIPIF1<0的直線(xiàn)l與圓C交于A，B兩點(diǎn)，則SKIPIF1<0的最小值是（

）．A．2 B．4 C．SKIPIF1<0 D．SKIPIF1<08．（2023·福建泉州·泉州五中校考模擬預(yù)測(cè)）已知復(fù)數(shù)SKIPIF1<0滿(mǎn)足SKIPIF1<0，則SKIPIF1<0的最大值為（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．3二、多選題9．（2023·江蘇·高二假期作業(yè)）過(guò)點(diǎn)SKIPIF1<0與SKIPIF1<0且半徑為2的圓的方程可以為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0三、填空題10．（2023秋·浙江湖州·高二統(tǒng)考期末）已知直線(xiàn)SKIPIF1<0平分圓SKIPIF1<0且與SKIPIF1<0互相平行，則SKIPIF1<0的距離是__________.四、解答題11．（2023春·甘肅蘭州·高二統(tǒng)考期中）已知點(diǎn)SKIPIF1<0求：(1)過(guò)點(diǎn)A，B且周長(zhǎng)最小的圓的方程；(2)過(guò)點(diǎn)A，B且圓心在直線(xiàn)SKIPIF1<0上的圓的方程.12．（2023秋·高一單元測(cè)試）已知圓SKIPIF1<0的圓心在SKIPIF1<0軸上，并且過(guò)SKIPIF1<0，SKIPIF1<0兩點(diǎn).(1)求圓SKIPIF1<0的方程；(2)若SKIPIF1<0為圓SKIPIF1<0上任意一點(diǎn)，定點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0滿(mǎn)足SKIPIF1<0，求點(diǎn)SKIPIF1<0的軌跡方程.B能力提升1