2025年高考数学圆锥曲线题型实战演练试卷

2025年高考数学圆锥曲线题型实战演练试卷考试时间：______分钟总分：______分姓名：______一、选择题（本大题共5小题，每小题5分，共25分。在每小题给出的四个选项中，只有一项是符合题目要求的。）1.已知F1，F2是椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点，P为椭圆上异于短轴端点的一点，且|PF1|=2|PF2|，则椭圆的离心率为（）A.1/2B.sqrt(3)/2C.1/sqrt(2)D.sqrt(2)/22.双曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的左焦点为F，右顶点为A，设P为双曲线上一点，且|PF|=2a，则∠APF的取值范围是（）A.(0,pi/2)B.(pi/2,pi)C.(pi/2,3pi/4)D.(3pi/4,pi)3.抛物线y^2=2px(p>0)的焦点到其准线的距离是（）A.p/2B.pC.2pD.4p4.已知点A(1,0)，F为抛物线y^2=4x的焦点，点P为抛物线上一点，则|PA|+|PF|的最小值为（）A.1B.2C.3D.45.直线y=kx与椭圆x^2+2y^2=2相交于M，N两点，若|MN|=sqrt(3)，则k的值为（）A.±1B.±sqrt(2)/2C.±sqrt(3)/3D.±sqrt(3)二、填空题（本大题共5小题，每小题5分，共25分。）6.椭圆x^2/9+y^2/4=1的焦点坐标是________。7.双曲线x^2/16-y^2/9=1的渐近线方程是________。8.抛物线y^2=8x的焦点到准线的距离是________。9.过抛物线y^2=2px(p>0)的焦点作一条弦，使其所在直线的倾斜角为45°，则弦长为________。10.已知F1，F2是椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点，P为椭圆上一点，且|PF1|+|PF2|=2a，则点P的轨迹方程是________。三、解答题（本大题共5小题，共50分。解答应写出文字说明、证明过程或演算步骤。）11.(本小题满分10分)已知椭圆C:x^2/25+y^2/16=1的左、右焦点分别为F1，F2，点P为椭圆C上一点，且|PF1|=3|PF2|，求点P的坐标。12.(本小题满分10分)已知双曲线C:x^2/9-y^2/16=1的左、右焦点分别为F1，F2，点P为双曲线C上一点，且|PF1|-|PF2|=2，求∠F1PF2的大小。13.(本小题满分10分)过抛物线y^2=8x的焦点F作一条直线，与抛物线交于A，B两点，若|AB|=8，求直线AB的方程。14.(本小题满分10分)已知椭圆C:x^2/25+y^2/16=1，直线l:y=kx+1与椭圆C相交于M，N两点。(1)求k的取值范围；(2)若|MN|=sqrt(10)，求k的值。15.(本小题满分10分)已知F1，F2是椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点，点P为椭圆上一点，且|PF1|+|PF2|=2a，设点A(2,0)。(1)求椭圆C的离心率e；(2)若|PA|=|PF2|，求椭圆C的方程。试卷答案一、选择题1.A2.C3.B4.B5.C二、填空题6.(±sqrt(5),0)7.y=±3/4x8.49.4p10.x^2/a^2+y^2/b^2=1三、解答题11.解析：由椭圆定义，|PF1|+|PF2|=2a。由|PF1|=3|PF2|，得|PF2|=a，|PF1|=3a。因为|PF1|+|PF2|=2a，所以a+3a=2a，即4a=2a，解得a=0，矛盾。说明点P不可能在椭圆上。重新理解题意：|PF1|=3|PF2|，且|PF1|+|PF2|=2a。设|PF2|=t，则|PF1|=3t，3t+t=2a，4t=2a，t=a/2。所以|PF2|=a/2，|PF1|=3a/2。椭圆半焦距c=sqrt(a^2-b^2)，c=sqrt(25-16)=3。F1(-3,0)，F2(3,0)。点P在x轴右侧（因为|PF1|>|PF2|），设P(x,y)。|PF1|=sqrt((x+3)^2+y^2)=3a/2=3*5/2=15/2。|PF2|=sqrt((x-3)^2+y^2)=a/2=5/2。两式平方：(x+3)^2+y^2=(15/2)^2=225/4(x-3)^2+y^2=(5/2)^2=25/4两式相减：(x+3)^2-(x-3)^2=225/4-25/4(x^2+6x+9)-(x^2-6x+9)=200/412x=50x=50/12=25/6。将x=25/6代入(x+3)^2+y^2=225/4：(25/6+3)^2+y^2=225/4(37/6)^2+y^2=225/41369/36+y^2=2025/4y^2=2025/4-1369/36y^2=2025*9/36-1369/36y^2=(18225-1369)/36y^2=16856/36=4214/9y=±sqrt(4214/9)=±sqrt(4214)/3。点P的坐标为(25/6,±sqrt(4214)/3)。12.解析：双曲线定义||PF1|-|PF2||=2a。由|PF1|-|PF2|=2，得||PF1|-|PF2||=2=2*3=6。双曲线a=3，c=5，离心率e=c/a=5/3。F1(-5,0)，F2(5,0)。点P在双曲线右支（因为|PF1|-|PF2|=2>0），设P(x,y)。|PF1|=sqrt((x+5)^2+y^2)，|PF2|=sqrt((x-5)^2+y^2)。|sqrt((x+5)^2+y^2)-sqrt((x-5)^2+y^2)|=6(x+5)^2+y^2-2*sqrt((x+5)^2+y^2)*sqrt((x-5)^2+y^2)+(x-5)^2+y^2=362x^2+2y^2+50-2*sqrt((x^2+10x+25+y^2)(x^2-10x+25+y^2))=36x^2+y^2+25-sqrt((x^2+y^2+25+10x)(x^2+y^2+25-10x))=18x^2+y^2+7=sqrt((x^2+y^2+25+10x)(x^2+y^2+25-10x))(x^2+y^2+7)^2=(x^2+y^2+25+10x)(x^2+y^2+25-10x)x^4+2x^2y^2+y^4+14x^2+14y^2+49=(x^2+y^2+25)^2-(10x)^2x^4+2x^2y^2+y^4+14x^2+14y^2+49=x^4+2x^2y^2+y^4+50x^2+50y^2+625-100x^2x^4+2x^2y^2+y^4+14x^2+14y^2+49=x^4+2x^2y^2+y^4-50x^2+50y^2+62564x^2-36y^2=57616x^2-9y^2=144x^2/9-y^2/16=1。点P在双曲线上。设直线MN的方程为y=m(x-5)。点M(x1,y1)，N(x2,y2)。联立方程组：y=m(x-5)x^2/9-y^2/16=1x^2/9-m^2(x-5)^2/16=116x^2-9m^2(x^2-10x+25)=144(16-9m^2)x^2+90m^2x-225m^2-144=0。x1+x2=-90m^2/(16-9m^2)，x1x2=-(225m^2+144)/(16-9m^2)。y1+y2=m(x1-5)+m(x2-5)=m(x1+x2-10)=m[-90m^2/(16-9m^2)-10]。y1y2=m^2(x1-5)(x2-5)=m^2[x1x2-5(x1+x2)+25]=m^2[-(225m^2+144)/(16-9m^2)-5(-90m^2/(16-9m^2)-10)+25]=m^2[-(225m^2+144)/(16-9m^2)+450m^2/(16-9m^2)+50+25]=m^2[225m^2/(16-9m^2)+75]=225m^4/(16-9m^2)+75m^2。|MN|=sqrt(1+m^2)|y1-y2|=sqrt(1+m^2)sqrt[(y1+y2)^2-4y1y2]=sqrt(1+m^2)sqrt{[m(-90m^2/(16-9m^2)-10)]^2-4[225m^4/(16-9m^2)+75m^2]}=sqrt(1+m^2)sqrt{m^2[(-90m^2-10(16-9m^2))^2/(16-9m^2)^2]-900m^4/(16-9m^2)-300m^2}=sqrt(1+m^2)sqrt{m^2[(-90m^2-160+90m^2)^2/(16-9m^2)^2]-900m^4/(16-9m^2)-300m^2}=sqrt(1+m^2)sqrt{m^2[-160^2/(16-9m^2)^2]-900m^4/(16-9m^2)-300m^2}=sqrt(1+m^2)sqrt{-25600m^2/(16-9m^2)^2-900m^4/(16-9m^2)-300m^2}=sqrt(1+m^2)sqrt{-900m^4-25600m^2(16-9m^2)/(16-9m^2)^2-300m^2(16-9m^2)^2/(16-9m^2)^2}=sqrt(1+m^2)sqrt{-900m^4-[25600*16m^2-25600*9m^4]/(16-9m^2)^2-[300*16^2-300*2*16*9m^2+300*81m^4]/(16-9m^2)^2}=sqrt(1+m^2)sqrt{-900m^4-[409600m^2-230400m^4]/(16-9m^2)^2-[76800-8640m^2+24300m^4]/(16-9m^2)^2}=sqrt(1+m^2)sqrt{(-230400m^4-24300m^4+230400m^4)/(16-9m^2)^2+8640m^2/(16-9m^2)^2-76800/(16-9m^2)^2}=sqrt(1+m^2)sqrt{(-24300m^4)/(16-9m^2)^2+8640m^2/(16-9m^2)^2-76800/(16-9m^2)^2}=sqrt(1+m^2)sqrt{[-24300m^4+8640m^2-76800]/(16-9m^2)^2}=sqrt(1+m^2)sqrt{[-24300(m^4-8640m^2/24300+76800/24300)]/(16-9m^2)^2}=sqrt(1+m^2)sqrt{[-24300(m^4-288m^2/81+256/81)]/(16-9m^2)^2}=sqrt(1+m^2)sqrt{[-24300(m^2-16/9)^2]/(16-9m^2)^2}(因为8640/24300=288/81=16/9)=sqrt(1+m^2)sqrt{[-24300(m^2-16/9)^2]/(9(16/9-m^2))^2}=sqrt(1+m^2)sqrt{(-24300(m^2-16/9)^2)/[9(16/9-m^2)^2]}=sqrt(1+m^2)sqrt{(-24300/9)*[(m^2-16/9)^2/(16/9-m^2)^2]}=sqrt(1+m^2)sqrt{(-2700)*[(m^2-16/9)^2/(16/9-m^2)^2]}=sqrt(1+m^2)sqrt{(-2700)*[(m^2-16/9)/(16/9-m^2)]^2}=sqrt(1+m^2)sqrt{2700*[(m^2-16/9)/(m^2-16/9)]^2}(因为m^2-16/9<0,so16/9-m^2>0)=sqrt(1+m^2)sqrt{2700*[-(16/9-m^2)]^2}=sqrt(1+m^2)*30*|16/9-m^2|=30sqrt(1+m^2)(16/9-m^2)。题目条件|MN|=sqrt(10)。30sqrt(1+m^2)(16/9-m^2)=sqrt(10)900(1+m^2)(16/9-m^2)=10100(1+m^2)(16-9m^2)=1010(1+m^2)(16-9m^2)=1160+160m^2-90m^4-9m^2=1-90m^4+151m^2+159=090m^4-151m^2-159=090(m^4-151m^2/90-159/90)=090(m^4-151/90m^2-53/30)=090(m^2)^2-151/90*2m^2-53/30=090(m^2)^2-302/90m^2-53/30=090(m^2)^2-151/45m^2-53/30=090*30*(m^2)^2-151*2*m^2-53*45=02700(m^2)^2-302m^2-2385=0(m^2)=[-(-302)±sqrt((-302)^2-4*2700*(-2385))]/(2*2700)=[302±sqrt(91204+25620000)]/5400=[302±sqrt(25631104)]/5400=[302±5062]/5400m^2=(302+5062)/5400=5364/5400=2682/2700=1341/1350或m^2=(302-5062)/5400=-4760/5400=-238/270=-119/135。m^2不能为负，所以m^2=1341/1350。m=±sqrt(1341/1350)。(2)椭圆方程：x^2/9+y^2/16=1。将m=sqrt(1341/1350)代入y=m(x-5)：y=sqrt(1341/1350)(x-5)。x^2/9+(sqrt(1341/1350)(x-5))^2/16=1x^2/9+(1341/1350)(x^2-10x+25)/16=1x^2/9+(1341/21600)(x^2-10x+25)=12400x^2+1341(x^2-10x+25)=216002400x^2+1341x^2-13410x+33525=216003741x^2-13410x+11925=0。Δ=(-13410)^2-4*3741*11925=179828100-179828100=0。x=-(-13410)/(2*3741)=13410/7482=5*2682/(2*3741)=5*1341/3741=5/7。y=sqrt(1341/1350)(5/7-5)=5*sqrt(1341/1350)*(-30/7)=-150*sqrt(1341/1350)/7。椭圆方程为：(x-5/7)^2/9+(y+150*sqrt(1341/1350)/7)^2/16=1。将m=-sqrt(1341/1350)代入y=m(x-5)：y=-sqrt(1341/1350)(x-5)。x^2/9+(-sqrt(1341/1350)(x-5))^2/16=1x^2/9+(1341/1350)(x^2-10x+25)/16=1x^2/9+(1341/21600)(x^2-10x+25)=12400x^2+1341(x^2-10x+25)=216002400x^2+1341x^2-13410x+33525=216003741x^2-13410x+11925=0。Δ=0。x=5/7。y=-sqrt(1341/1350)(5/7-5)=-5*sqrt(1341/1350)*(-30/7)=150*sqrt(1341/1350)/7。椭圆方程为：(x-5/7)^2/9+(y-150*sqrt(1341/1350)/7)^2/16=1。综上所述，椭圆方程为：(x-5/7)^2/9+(y±150*sqrt(1341/1350)/7)^2/16=1。15.解析：(1)由题意，点P在椭圆上，所以|PF1|+|PF2|=2a。椭圆的离心率e=c/a=sqrt(1-(b^2/a^2))。由题意，|PF1|+|PF2|=2a，所以c/a=sqrt(1-(b^2/a^2))=sqrt(1-(1/4))=sqrt(3/4)=sqrt(3)/2。所以椭圆的离心率e=sqrt(3)/2。(2)设F1(-c,0)，F2(c,0)，A(2,0)。由(1)知，e=c/a=sqrt(3)/2，所以c=asqrt(3)/2。a^2=b^2+c^2，a^2=1/4+(a^2/4)*3/4，a^2=1/4+3a^2/16，16a^2=4+3a^2，13a^2=4，a^2=4/13。b^2=a^2-c^2=4/13-(4/13*3/4)=4/13-3/13=1/13。椭圆方程为：x^2/(4/13)+y^2/(1/13)=1，即13x^2/4+13y^2/1=1，x^2/4+y^2=1/13。由|PA|=|PF2|，得sqrt((x-2)^2+y^2)=sqrt((x-c)^2+y^2)。(x-2)^2+y^2=(x-c)^2+y^2。(x-2)^2=(x-c)^2。