2026年济南高三数学高考三模冲刺卷：圆锥曲线定点定值专题（校际联考版第5套）含参考答案、逐题解析与评分细则

济南高三数学高考三模冲刺·圆锥曲线定点定值专题·校际联考版第5套满分150分考试时间120分钟2026年济南高三数学高考三模冲刺卷：圆锥曲线定点定值专题（校际联考版第5套）含参考答案、逐题解析与评分细则济南市校际联考版·2026届高三数学高考三模冲刺卷（第5套）专题定位：圆锥曲线中的定点、定直线、定值与切点弦问题满分：150分考试时间：120分钟注意事项：1.答题前请将学校、班级、姓名、考号填写在指定位置；2.选择题须把唯一正确选项或全部正确选项填入答题栏，多项选择题少选且无错选按规则给分；3.填空题只需写出最后结果，解答题须写出必要推理、计算过程和结论；4.保持卷面整洁，所有答案写在指定位置。学校班级姓名考号学生答题栏题号12345678答案得分多项选择题答题栏（全部选对得5分；少选且无错选得2分；错选、多选或不选得0分）：题号9101112答案填空题作答区：13.____________14.____________15.____________16.____________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）已知椭圆E的方程为x²/16＋y²/7＝1，左、右焦点分别为F₁，F₂，则该椭圆的离心率为（）A.1/2B.3/4C.√7/4D.4/32.（5分）抛物线C：y²＝8x的焦点为F，点P在C上且横坐标为3，则|PF|＝（）A.3B.4C.5D.63.（5分）双曲线H：x²/9－y²/16＝1的任意一条切线与两条渐近线分别交于M，N，则△OMN的面积恒为（）A.6B.9C.12D.244.（5分）抛物线C：y²＝4x的焦点弦AB过焦点F，则1/|AF|＋1/|BF|的定值为（）A.1/2B.1C.2D.45.（5分）椭圆E：x²/9＋y²/4＝1中，动弦AB过点M(0，1)，A，B处切线交于T，则T所在的定直线为（）A.x＝9B.y＝4C.x＝0D.y＝16.（5分）双曲线H：x²/4－y²＝1中，动弦AB过点P(1，0)，A，B处切线交于T，则T所在的定直线为（）A.x＝4B.x＝1C.y＝4D.y＝17.（5分）椭圆E：x²/25＋y²/16＝1上一点P(3，16/5)处的切线方程为（）A.5x＋3y＝25B.3x－5y＝25C.5x－3y＝25D.3x＋5y＝258.（5分）已知双曲线焦点在x轴上，渐近线斜率为±√3/3，半焦距c＝4，则双曲线方程可写为（）A.x²/4－y²/12＝1B.x²/4－y²/3＝1C.x²/12－y²/4＝1D.x²/16－y²/4＝1二、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，少选且无错选得2分，错选、多选或不选得0分。9.（5分）已知椭圆E：x²/16＋y²/7＝1，下列结论正确的是（）A.焦点坐标为(－3，0)，(3，0)B.离心率e＝3/4C.通径长为7/2D.任意点P∈E满足|PF₁|＋|PF₂|＝1610.（5分）抛物线C：y²＝12x的焦点为F，过F的直线交C于A，B。若A，B对应参数分别为t₁，t₂，则下列结论正确的是（）A.t₁t₂＝－1B.|AF|·|BF|的最小值为36C.1/|AF|＋1/|BF|＝1/3D.A，B处切线交点恒在直线x＝3上11.（5分）双曲线H：x²/4－y²/12＝1的下列说法正确的是（）A.离心率为2B.渐近线方程为y＝±√3xC.焦点坐标为(－4，0)，(4，0)D.实轴长为212.（5分）椭圆E：x²/9＋y²/4＝1，点P(6，t)在直线x＝6上，由P作E的两条切线，切点为A，B。下列结论正确的是（）A.切点弦AB的方程为2x/3＋ty/4＝1B.AB恒过点(3/2，0)C.当t＝2时，AB的方程为4x＋3y＝6D.当t＝0时，两条切线重合三、填空题：本大题共4小题，每小题5分，共20分。13.（5分）椭圆E：x²/25＋y²/16＝1的右焦点为F，过F的动直线交E于A，B，则1/|AF|＋1/|BF|的定值为________。作答：________________________________________________14.（5分）抛物线C：y²＝8x的焦点弦AB过焦点F，则1/|AF|＋1/|BF|＝________。作答：________________________________________________15.（5分）椭圆E：x²/9＋y²/4＝1中，动弦AB过点M(1，1)，A，B处切线交于T，则T所在定直线的一般式为________。作答：________________________________________________16.（5分）双曲线H：x²/16－y²/9＝1的任意一条切线与两条渐近线围成的、以原点为顶点的三角形面积恒为________。作答：________________________________________________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E：x²/9＋y²/4＝1，点M(0，1)。动直线l过M且与E交于A，B两点，A，B两点处的切线交于T。

（1）证明：点T在一条定直线上；

（2）求|OT|的最小值，并说明何时取得。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（12分）已知抛物线C：y²＝4x，焦点为F。过F的动直线与C交于A，B两点，A，B处的切线交于T。

（1）用参数法证明T在定直线x＝－1上；

（2）证明1/|AF|＋1/|BF|为定值，并求该定值。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（12分）已知双曲线H：x²/4－y²＝1，P(x₀，y₀)是第一象限内H上一点，过P作H的切线，该切线与两条渐近线分别交于M，N。

（1）证明△OMN的面积为定值，并求该值；

（2）若该切线斜率为1，求点P坐标及切线方程。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（12分）已知椭圆E：x²/9＋y²/4＝1，右焦点为F。过F的动直线与E交于A，B两点。

（1）证明1/|AF|＋1/|BF|为定值，并求该定值；

（2）求焦点弦AB长度的最小值，并写出此时直线AB的方向。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（12分）已知抛物线C：y²＝4x，点T(2，m)在直线x＝2上。过T作C的两条切线，切点分别为A，B。

（1）求切点弦AB的方程，并证明AB过定点；

（2）当m＝4时，求A，B的坐标。作答区：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________