新高考數(shù)學(xué)一輪復(fù)習(xí)精講精練10.4 雙曲線（基礎(chǔ)版）（原卷版）

10.4雙曲線（精講）（基礎(chǔ)版）思維導(dǎo)圖思維導(dǎo)圖考點(diǎn)呈現(xiàn)考點(diǎn)呈現(xiàn)例題剖析例題剖析考點(diǎn)一雙曲線的定義及應(yīng)用【例1-1】（2022·潮州二模）若點(diǎn)P是雙曲線SKIPIF1<0上一點(diǎn)，SKIPIF1<0，SKIPIF1<0分別為SKIPIF1<0的左、右焦點(diǎn)，則“SKIPIF1<0”是“SKIPIF1<0”的（）．A．充分不必要條件 B．必要不充分條件C．充要條件 D．既不充分也不必要條件【例1-2】（2022·成都模擬）設(shè)SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左，右焦點(diǎn)，點(diǎn)P在雙曲線C的右支上，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0面積為（）．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【例1-3】（2022常州期中）已知雙曲線SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，SKIPIF1<0為雙曲線左支上一點(diǎn)，點(diǎn)SKIPIF1<0，則SKIPIF1<0周長(zhǎng)的最小值為（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【例1-4】（2021河北月考）已知方程SKIPIF1<0表示雙曲線，則實(shí)數(shù)SKIPIF1<0的取值范圍為（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0【一隅三反】1．（2022高三上·廣東開學(xué)考）“k<2”是“方程SKIPIF1<0表示雙曲線”的（）A．充分不必要條件 B．必要不充分條件C．充要條件 D．既不充分也不必要條件2．（2022·運(yùn)城模擬）已知雙曲線SKIPIF1<0的左右焦點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是雙曲線上一點(diǎn)，SKIPIF1<0，則SKIPIF1<0（）A．1或13 B．1 C．13 D．93．（2022紅塔月考）已知SKIPIF1<0是雙曲線SKIPIF1<0的左焦點(diǎn)，點(diǎn)SKIPIF1<0，SKIPIF1<0是雙曲線右支上的動(dòng)點(diǎn)，則SKIPIF1<0的最小值為（）A．9 B．5 C．8 D．44（2022廣東）已知SKIPIF1<0，SKIPIF1<0分別是雙曲線SKIPIF1<0的左?右焦點(diǎn)，若P是雙曲線左支上的點(diǎn)，且SKIPIF1<0.則SKIPIF1<0的面積為（）A．8 B．SKIPIF1<0 C．16 D．SKIPIF1<0考點(diǎn)二雙曲線的離心率及漸近線【例2-1】（2022龍崗期中）雙曲線SKIPIF1<0的漸近線方程是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【例2-2】（2022長(zhǎng)春月考）在SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0.若以A，B為焦點(diǎn)的雙曲線經(jīng)過點(diǎn)C，則該雙曲線的離心率為()A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【例2-3】（2022·重慶市模擬）已知雙曲線C：SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0在SKIPIF1<0軸上，SKIPIF1<0為等邊三角形，且線段SKIPIF1<0的中點(diǎn)恰在雙曲線C上，則雙曲線C的離心率為（）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<0【一隅三反】1．（2022·湖南模擬）已知O是坐標(biāo)原點(diǎn)，F(xiàn)是雙曲線SKIPIF1<0的右焦點(diǎn)，過雙曲線C的右頂點(diǎn)且垂直于x軸的直線與雙曲線C的一條漸近線交于A點(diǎn)，若以F為圓心的圓經(jīng)過點(diǎn)A，O，則雙曲線C的漸近線方程為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022高三上·廣西開學(xué)考）已知SKIPIF1<0，SKIPIF1<0是雙曲線C的兩個(gè)焦點(diǎn)，P為雙曲線上的一點(diǎn)，且SKIPIF1<0；則C的離心率為（）A．1 B．2 C．3 D．43．（202懷仁期末）設(shè)SKIPIF1<0，SKIPIF1<0分別是雙曲線SKIPIF1<0的左、右焦點(diǎn)，若雙曲線右支上存在一點(diǎn)SKIPIF1<0，使SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)），且SKIPIF1<0，則雙曲線的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04（2022德州月考）已知雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，曲線SKIPIF1<0上一點(diǎn)SKIPIF1<0到SKIPIF1<0軸的距離為SKIPIF1<0，且SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2022遼寧期中）（多選）已知雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是雙曲線上一點(diǎn)，若SKIPIF1<0，則該雙曲線的離心率可以是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．2考點(diǎn)三雙曲線的標(biāo)準(zhǔn)方程【例3-1】（2022·海寧模擬）已知雙曲線C的漸近線方程為SKIPIF1<0，且焦距為10，則雙曲線C的標(biāo)準(zhǔn)方程是（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0【例3-2】（2022·河西模擬）已知雙曲線的一個(gè)焦點(diǎn)與拋物線SKIPIF1<0的焦點(diǎn)重合，且雙曲線上的一點(diǎn)SKIPIF1<0到雙曲線的兩個(gè)焦點(diǎn)的距離之差的絕對(duì)值等于6，則雙曲線的標(biāo)準(zhǔn)方程為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【一隅三反】1．（2022·河南模擬）已知雙曲線SKIPIF1<0的一條漸近線過點(diǎn)SKIPIF1<0，SKIPIF1<0是SKIPIF1<0的左焦點(diǎn)，且SKIPIF1<0，則雙曲線SKIPIF1<0的方程為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02（2021高三上·寧波期末）已知雙曲線SKIPIF1<0與雙曲線SKIPIF1<0有相同的漸近線，且它們的離心率不相同，則下列方程中有可能為雙曲線SKIPIF1<0的標(biāo)準(zhǔn)方程的是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03（2022南山期末）已知雙曲線C過點(diǎn)SKIPIF1<0且漸近線為SKIPIF1<0，則雙曲線C的方程是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022商丘）若方程SKIPIF1<0表示雙曲線，則實(shí)數(shù)m的取值范圍是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2021肇東月考）以SKIPIF1<0，SKIPIF1<0為焦點(diǎn)且過點(diǎn)SKIPIF1<0的雙曲線方程是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0考點(diǎn)四直線與雙曲線的位置關(guān)系【例4-1】（2022·全國(guó)·高三專題練習(xí)）過SKIPIF1<0且與雙曲線SKIPIF1<0有且只有一個(gè)公共點(diǎn)的直線有（

）A．1條 B．2條 C．3條 D．4條【例4-2】（2022·全國(guó)·專題練習(xí)）若過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與雙曲線SKIPIF1<0:SKIPIF1<0的右支相交于不同兩點(diǎn)，則直線SKIPIF1<0斜率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【一隅三反】1．（2022·安徽）直線SKIPIF1<0與雙曲線SKIPIF1<0沒有公共點(diǎn)，則斜率k的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2022·全國(guó)·高三專題練習(xí)）若雙曲線SKIPIF1<0的一個(gè)頂點(diǎn)為A，過點(diǎn)A的直線SKIPIF1<0與雙曲線只有一個(gè)公共點(diǎn)，則該雙曲線的焦距為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0考點(diǎn)五弦長(zhǎng)與中點(diǎn)弦【例5-1】（2021·江西?。┮阎p曲線x2－y2＝a2（a＞0）與直線y＝SKIPIF1<0x交于A、B兩點(diǎn)，且｜AB｜＝2SKIPIF1<0，則a＝_____【例5-2】（2022·河南·模擬預(yù)測(cè)（文））已知雙曲線SKIPIF1<0的離心率為SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，則SKIPIF1<0與SKIPIF1<0的斜率的乘積為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【一隅三反】1．（2021·全國(guó)·高二課時(shí)練習(xí)）已知雙曲線SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線l與雙曲線C交于M?N兩點(diǎn)，若P為線段MN的中點(diǎn)，則弦長(zhǎng)|MN|等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·全國(guó)·高三專題練習(xí)）已知雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0過左焦點(diǎn)SKIPIF1<0作斜率為2的直線與雙曲線交于A，B兩點(diǎn)，P是AB的中點(diǎn)，O為坐標(biāo)原點(diǎn)，若直線OP的斜率為SKIPIF1<0，則b的值是（

）A．2 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·山東煙臺(tái)·三模）過雙曲線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的焦點(diǎn)且斜率不為0的直線交SKIPIF1<0于A，SKIPIF1<0兩點(diǎn)，SKIPIF1<0為SKIPIF1<0中點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<04．（2023·全國(guó)·高三專題練習(xí)）已知雙曲線C的中心在坐標(biāo)原點(diǎn)，其中一個(gè)焦點(diǎn)為SKIPIF1<0，過F的直線l與雙曲線C交于A、B兩點(diǎn)，且AB的中點(diǎn)為SKIPIF1<0，則C的離心率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2022·四川?。┮阎行脑谠c(diǎn)的雙曲線C的右焦點(diǎn)為（2，0），右頂點(diǎn)為（SKIPIF1<0，0）．(1)求雙曲線C的方程；(2)若直線l：y＝x+2與雙曲線交于A，B兩點(diǎn)，求弦長(zhǎng)|AB|．10.4雙曲線（精練）（基礎(chǔ)版）題組一題組一雙曲線的定義及應(yīng)用1．（2021·太原期末）已知SKIPIF1<0，SKIPIF1<0分別是雙曲線SKIPIF1<0的左右焦點(diǎn)，點(diǎn)P在該雙曲線上，若SKIPIF1<0，則SKIPIF1<0（）A．4 B．4或6 C．3 D．3或72．（2022郫都期中）雙曲線SKIPIF1<0的兩個(gè)焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，雙曲線上一點(diǎn)SKIPIF1<0到SKIPIF1<0的距離為11，則點(diǎn)SKIPIF1<0到SKIPIF1<0的距離為（）A．1 B．21 C．1或21 D．2或213．（2021懷仁期中）已知SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左右焦點(diǎn)，過SKIPIF1<0的直線SKIPIF1<0與曲線SKIPIF1<0的右支交于SKIPIF1<0兩點(diǎn)，則SKIPIF1<0的周長(zhǎng)的最小值為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022奉賢期中）已知SKIPIF1<0是雙曲線SKIPIF1<0右支上的一點(diǎn)，雙曲線的一條漸近線方程為SKIPIF1<0.設(shè)SKIPIF1<0分別為雙曲線的左、右焦點(diǎn).若SKIPIF1<0，則SKIPIF1<0.5．（2022·開封模擬）若雙曲線SKIPIF1<0的焦距為SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0．6．（2022·岳普湖模擬）已知雙曲線SKIPIF1<0，F(xiàn)1，F(xiàn)2是雙曲線的左右兩個(gè)焦點(diǎn)，P在雙曲線上且在第一象限，圓M是△F1PF2的內(nèi)切圓.則M的橫坐標(biāo)為，若F1到圓M上點(diǎn)的最大距離為SKIPIF1<0，則△F1PF2的面積為.7．（2021溫州期中）已知雙曲線x2-y2=1，點(diǎn)F1，F(xiàn)2為其兩個(gè)焦點(diǎn)，點(diǎn)P為雙曲線上一點(diǎn)，若PF1⊥PF2，則∣PF1∣+∣PF2∣的值為.題組二題組二雙曲線的離心率及漸近線1．（2021高三上·南開期末）已知雙曲線SKIPIF1<0，過原點(diǎn)作一條傾斜角為SKIPIF1<0的直線分別交雙曲線左、右兩支于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，以線段SKIPIF1<0為直徑的圓過右焦點(diǎn)SKIPIF1<0，則雙曲線的離心率為（）.A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022湖南月考）已知雙曲線的左焦點(diǎn)為SKIPIF1<0，右焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為雙曲線右支上一點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，滿足SKIPIF1<0，且SKIPIF1<0，則該雙曲線的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．SKIPIF1<03．（2021·全國(guó)甲卷）已知F1，F(xiàn)2是雙曲線C的兩個(gè)焦點(diǎn)，P為C上一點(diǎn)，且∠F1PF2=60°，|PF1|=3|PF2|，則C的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022·靖遠(yuǎn)模擬）若雙曲線SKIPIF1<0的兩條漸近線與直線y＝2圍成了一個(gè)等邊三角形，則C的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．25．（2022·新鄉(xiāng)三模）已知雙曲線SKIPIF1<0的頂點(diǎn)到一條漸近線的距離為實(shí)軸長(zhǎng)的SKIPIF1<0，則雙曲線C的離心率為（）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．36．（2022·湘贛皖模擬）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，雙曲線C上一點(diǎn)P到x軸的距離為c，且SKIPIF1<0，則雙曲線C的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．（2022·濟(jì)南二模）已知SKIPIF1<0，SKIPIF1<0分別為雙曲線SKIPIF1<0的左?右焦點(diǎn)，點(diǎn)P在雙曲線上，若SKIPIF1<0，SKIPIF1<0，則雙曲線的離心率為.8．（2022·汝州模擬）已知雙曲線SKIPIF1<0的兩條漸近線所夾銳角為SKIPIF1<0，則雙曲線的離心率為．題組三題組三雙曲線的標(biāo)準(zhǔn)方程1．（2022·安徽模擬）與橢圓SKIPIF1<0共焦點(diǎn)且過點(diǎn)SKIPIF1<0的雙曲線的標(biāo)準(zhǔn)方程為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022合肥期末）已知點(diǎn)SKIPIF1<0分別是等軸雙曲線SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上，SKIPIF1<0，SKIPIF1<0的面積為8，則雙曲線SKIPIF1<0的方程為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022資陽(yáng)期末）已知雙曲線SKIPIF1<0過三點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0中的兩點(diǎn)，則SKIPIF1<0的方程為.4．（2022徐匯期末）已知雙曲線經(jīng)過點(diǎn)SKIPIF1<0，其漸近線方程為SKIPIF1<0，則該雙曲線的方程為．5．（2022河南月考）經(jīng)過點(diǎn)SKIPIF1<0且與雙曲線SKIPIF1<0有公共漸近線的雙曲線方程為．6．（2022·湖北模擬）在平面直角坐標(biāo)系中，已知圓SKIPIF1<0：SKIPIF1<0，點(diǎn)SKIPIF1<0，SKIPIF1<0是圓SKIPIF1<0上任意一點(diǎn)，線段SKIPIF1<0的垂直平分線與直線SKIPIF1<0相交于點(diǎn)SKIPIF1<0，設(shè)點(diǎn)SKIPIF1<0的軌跡為曲線SKIPIF1<0，則曲線SKIPIF1<0的方程為.7．（2022·遼寧模擬）已知雙曲線SKIPIF1<0的一條漸近線方程為SKIPIF1<0，一個(gè)焦點(diǎn)到一條漸近線的距離為SKIPIF1<0，則雙曲線的標(biāo)準(zhǔn)方程為.8．（2022·寧德模擬）若過點(diǎn)SKIPIF1<0的雙曲線的漸近線為SKIPIF1<0，則該雙曲線的標(biāo)準(zhǔn)方程是.9．（2022·廣州模擬）寫出一個(gè)同時(shí)滿足下列性質(zhì)①②③的雙曲線方程．①中心在原點(diǎn)，焦點(diǎn)在y軸上；②一條漸近線方程為SKIPIF1<0﹔③焦距大于10題組四題組四直線與雙曲線的位置關(guān)系1．（2022·全國(guó)·課時(shí)練習(xí)）已知直線l的方程為SKIPIF1<0，雙曲線C的方程為SKIPIF1<0.若直線l與雙曲線C的右支相交于不同的兩點(diǎn)，則實(shí)數(shù)k的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·全國(guó)·課時(shí)練習(xí)）直線SKIPIF1<0與雙曲線SKIPIF1<0上支的交點(diǎn)個(gè)數(shù)為______．3．（2022·全國(guó)·課時(shí)練習(xí)）直線SKIPIF1<0與雙曲線SKIPIF1<0的交點(diǎn)坐標(biāo)為______．4．（2022·全國(guó)·高三專題練習(xí)）直線SKIPIF1<0與雙曲線SKIPIF1<0沒有交點(diǎn)，則SK