2026年无锡高三数学高考三模冲刺卷：圆锥曲线定点定值专题（教师命题组版第3套）含参考答案、逐题解析与评分细则

2026年无锡高三数学高考三模冲刺卷：圆锥曲线定点定值专题（教师命题组版第3套）含参考答案、逐题解析与评分细则无锡市高三数学教师命题组版第3套2026届高考三模冲刺数学试卷专题定位：圆锥曲线定点、定值、切线、弦中点与极线思想综合训练地区/学校簇：无锡适用对象：高三满分：150分考试时间：120分钟注意事项1.本卷分为单项选择题、多项选择题、填空题和解答题四部分，共22题，满分150分。2.单项选择题每题只有一个正确选项；多项选择题至少有两个正确选项，全部选对得满分，漏选得部分分，有错选不得分。3.填空题只需写出最终结果；解答题必须写出必要的推理、计算过程和结论，答案写在指定作答区域内。4.书写应规范、清楚；涉及圆锥曲线参数、斜率、切线和定点定值结论时，要说明变量范围和关键等式。一、单项选择题：本大题共8小题，每小题5分，共40分。每小题只有一个选项符合题意。1.（5分）椭圆C:x²/4+y²/3=1的离心率为（）。A.1/3B.1/2C.√3/2D.3/42.（5分）抛物线C:y²=8x的焦点为F，点P(2,4)在C上，则PF的长为（）。A.4B.2C.6D.83.（5分）双曲线C:x²/9-y²/16=1的一条渐近线方程为（）。A.y=±3x/4B.y=±5x/3C.y=±4x/3D.y=±3x/54.（5分）已知椭圆C:x²/a²+y²/b²=1(a>b>0)，过右焦点且垂直于长轴的弦长为（）。A.b²/aB.4b²/aC.2abD.2b²/a5.（5分）抛物线C:y²=4x上参数为t=2的点处切线方程为（）。A.x-2y+4=0B.2x-y+4=0C.x+2y-4=0D.2x+y-4=06.（5分）椭圆C:x²/9+y²/4=1的一条弦以M(1,1)为中点，则该弦的斜率为（）。A.-9/4B.-4/9C.4/9D.9/47.（5分）双曲线C:x²/a²-y²/b²=1(a>0,b>0)的离心率为√2，则下列结论正确的是（）。A.b²=2a²B.a²=2b²C.a=bD.c=b8.（5分）抛物线C:y²=4x，过焦点F(1,0)的直线交C于A,B，A,B处切线交于T，则T所在的定直线为（）。A.x=-1B.x=1C.y=0D.y=1单项选择题答题栏：题号12345678答案二、多项选择题：本大题共4小题，每小题5分，共20分。全部选对得5分，漏选得2分，有错选得0分。9.（5分）关于椭圆C:x²/25+y²/16=1，下列说法正确的是（）。A.离心率为3/5B.焦点为(±3,0)C.点(3,16/5)在椭圆上D.在点(3,16/5)处的切线为3x/25+y/5=110.（5分）关于抛物线C:y²=8x，下列说法正确的是（）。A.焦点为(2,0)B.参数点可写成(2t²,4t)C.参数为t处的切线为ty=x+2t²D.准线方程为x=211.（5分）关于双曲线C:x²/4-y²/5=1，下列结论正确的是（）。A.离心率为3/2B.渐近线为y=±√5x/2C.点(4,√15)在双曲线上D.在点(3,5/2)处的切线为3x/4-y/2=112.（5分）抛物线C:y²=4x上两点A(t₁²,2t₁),B(t₂²,2t₂)满足t₁t₂=-1。下列结论正确的是（）。A.弦AB过定点(1,0)B.两点处切线交点在定直线x=-1上C.两条切线互相垂直D.弦AB的斜率为定值多项选择题答题栏：题号9101112答案三、填空题：本大题共4小题，每小题5分，共20分。请把答案填写在横线上。13.（5分）椭圆C:x²/9+y²/5=1的一条弦以M(1,1)为中点，则该弦斜率为__________。14.（5分）抛物线C:y²=6x中，过焦点且斜率为1的焦点弦长为__________。15.（5分）双曲线C:x²/a²-y²/b²=1的离心率为2，且过点(4,√3)，则其渐近线斜率的绝对值为__________。16.（5分）椭圆C:x²/4+y²=1，从点P(4,0)向椭圆引两条切线，若两切线斜率分别为k₁、k₂，则k₁k₂=__________。四、解答题：本大题共6小题，共70分。解答应写出文字说明、证明过程或演算步骤。17.（10分）已知椭圆C:x²/a²+y²/b²=1(a>b>0)的焦点为F₁(-2,0),F₂(2,0)，且过点P(3/2,√15/2)。（1）求椭圆C的方程；（2）若弦AB以M(1,1)为中点，求直线AB的方程；（3）设A,B处切线交于T，求T的坐标。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（12分）已知抛物线C:y²=4x，焦点为F。直线l过F，交C于不同两点A,B。（1）写出C上参数为t的点及该点处切线方程；（2）证明A,B处切线的交点T在一条定直线上；（3）求TF的最小值，并说明取等条件。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（12分）双曲线C:x²/a²-y²/b²=1(a>0,b>0)的离心率为3/2，且过点(4,√15)。（1）求C的方程；（2）求C在点Q(3,5/2)处的切线方程；（3）该切线交两条渐近线于M,N，证明Q为MN中点，并求△OMN面积的定值。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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参考答案与逐题解析、评分细则本部分供教师讲评、阅卷和学生订正使用。客观题给出关键依据；解答题按步骤列出采分点。一、单项选择题答案与解析1.答案：B。解析：椭圆中a²=4,b²=3，故c²=a²-b²=1，e=c/a=1/2。A、C、D均把c或a的关系误配。评分细则：选B得5分；选其他或不选得0分。2.答案：A。解析：抛物线y²=8x可写成y²=4ax，a=2，焦点F(2,0)。点P(2,4)到焦点距离为4，也等于到准线x=-2的距离。评分细则：选A得5分；选其他或不选得0分。3.答案：C。解析：双曲线x²/a²-y²/b²=1的渐近线为y=±(b/a)x。本题a=3,b=4，故y=±4x/3。评分细则：选C得5分；选其他或不选得0分。4.答案：D。解析：过焦点且垂直长轴的弦称为通径。令x=c，有y²/b²=1-c²/a²=b²/a²，弦长为2b²/a。评分细则：选D得5分；选其他或不选得0分。5.答案：A。解析：抛物线y²=4x的参数点为(t²,2t)，切线为ty=x+t²。取t=2，得2y=x+4，即x-2y+4=0。评分细则：选A得5分；选其他或不选得0分。6.答案：B。解析：椭圆x²/a²+y²/b²=1中点弦斜率公式为k=-b²x₀/(a²y₀)。取a²=9,b²=4,M(1,1)，得k=-4/9。评分细则：选B得5分；选其他或不选得0分。7.答案：C。解析：双曲线e²=c²/a²=(a²+b²)/a²=1+b²/a²。若e=√2，则b²/a²=1，所以a=b。评分细则：选C得5分；选其他或不选得0分。8.答案：A。解析：设A,B参数为t₁,t₂。过焦点(1,0)的弦满足t₁t₂=-1，两切线交点为(t₁t₂,t₁+t₂)，故横坐标恒为-1。评分细则：选A得5分；选其他或不选得0分。二、多项选择题答案与解析9.答案：ABCD。解析：椭圆中a=5,b=4,c=3，所以离心率e=3/5，焦点为(±3,0)。代入点(3,16/5)得9/25+256/(25·16)=1。切线公式xx₀/25+yy₀/16=1化为3x/25+y/5=1，四项均正确。评分细则：全部选对得5分，漏选且无错选得2分，有错选或不选得0分。10.答案：ABC。解析：由y²=8x=4ax知a=2，焦点为(2,0)，参数点为(2t²,4t)，切线为ty=x+2t²。准线应为x=-2，故D错误。评分细则：全部选对得5分，漏选且无错选得2分，有错选或不选得0分。11.答案：ABCD。解析：本题a²=4,b²=5,c²=9，故e=3/2，渐近线为y=±√5x/2。点(4,√15)满足16/4-15/5=1。点(3,5/2)在曲线上，切线为3x/4-(5/2)y/5=1，即3x/4-y/2=1。评分细则：全部选对得5分，漏选且无错选得2分，有错选或不选得0分。12.答案：ABC。解析：抛物线y²=4x弦AB方程为(t₁+t₂)y=2x+2t₁t₂。当t₁t₂=-1时，弦过(1,0)；两切线交点为(t₁t₂,t₁+t₂)，在x=-1上；切线斜率分别为1/t₁,1/t₂，乘积为-1，故垂直。弦斜率随t₁+t₂变化，D错误。评分细则：全部选对得5分，漏选且无错选得2分，有错选或不选得0分。三、填空题答案与解析13.答案：-5/9。解析：由中点弦斜率公式k=-b²x₀/(a²y₀)，取a²=9,b²=5,M(1,1)，得k=-5/9。评分细则：写出-5/9得5分；符号错误扣3分；只写公式未代值得2分。14.答案：12。解析：抛物线y²=6x中4a=6，故a=3/2。焦点弦与轴夹角为45°，弦长L=4a/sin²45°=6/(1/2)=12。评分细则：答案12得5分；方法正确但计算错得2分。15.答案：√3。解析：由e=2得c²=4a²，又c²=a²+b²，故b²