第15讲 抛物线（七大题型）（学生版）-2024年高中数学新高二暑期衔接讲义

第第页第15讲抛物线【题型归纳目录】题型一：抛物线的定义题型二：求抛物线的标准方程题型三：抛物线的综合问题题型四：轨迹方程题型五：抛物线的几何性质题型六：抛物线中的范围与最值问题题型七：焦半径问题【知识点梳理】知识点一：抛物线的定义定义：平面内与一个定点SKIPIF1<0和一条定直线SKIPIF1<0(SKIPIF1<0不经过点SKIPIF1<0)的距离相等的点的轨迹叫做抛物线，定点SKIPIF1<0叫做抛物线的焦点，定直线SKIPIF1<0叫做抛物线的准线．知识点诠释:(1)上述定义可归纳为“一动三定”，一个动点，一定直线；一个定值(2)定义中的隐含条件：焦点F不在准线SKIPIF1<0上，若F在SKIPIF1<0上，抛物线变为过F且垂直与SKIPIF1<0的一条直线.(3)抛物线定义建立了抛物线上的点、焦点、准线三者之间的距离关系，在解题时常与抛物线的定义联系起来，将抛物线上的动点到焦点的距离与动点到准线的距离互化，通过这种转化使问题简单化.知识点二：抛物线的标准方程抛物线标准方程的四种形式：根据抛物线焦点所在半轴的不同可得抛物线方程的的四种形式SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0SKIPIF1<0。知识点诠释：①只有当抛物线的顶点是原点，对称轴是坐标轴时，才能得到抛物线的标准方程；②抛物线的焦点在标准方程中一次项对应的坐标轴上，且开口方向与一次项的系数的正负一致，比如抛物线SKIPIF1<0的一次项为SKIPIF1<0，故其焦点在SKIPIF1<0轴上，且开口向负方向(向下)③抛物线标准方程中一次项的系数是焦点的对应坐标的4倍.④从方程形式看，求抛物线的标准方程仅需确定一次项系数。用待定系数法求抛物线的标准方程时，首先根据已知条件确定抛物线的标准方程的类型(一般需结合图形依据焦点的位置或开口方向定型)，然后求一次项的系数，否则，应展开相应的讨论.⑤在求抛物线方程时，由于标准方程有四种形式，易混淆，可先根据题目的条件作出草图，确定方程的形式，再求参数p，若不能确定是哪一种形式的标准方程，应写出四种形式的标准方程来，不要遗漏某一种情况。知识点三：抛物线的简单几何性质：抛物线标准方程SKIPIF1<0的几何性质范围：SKIPIF1<0，SKIPIF1<0，抛物线y2=2px(p＞0)在y轴的右侧，开口向右，这条抛物线上的任意一点M的坐标(x，y)的横坐标满足不等式x≥0；当x的值增大时，|y|也增大，这说明抛物线向右上方和右下方无限延伸。抛物线是无界曲线。对称性：关于x轴对称抛物线y2=2px(p＞0)关于x轴对称，我们把抛物线的对称轴叫做抛物线的轴。抛物线只有一条对称轴。顶点：坐标原点抛物线y2=2px(p＞0)和它的轴的交点叫做抛物线的顶点。抛物线的顶点坐标是(0，0)。抛物线标准方程几何性质的对比图形标准方程y2=2px(p＞0)y2=－2px(p＞0)x2=2py(p＞0)x2=－2py(p＞0)顶点O(0，0)范围x≥0，SKIPIF1<0x≤0，SKIPIF1<0y≥0，SKIPIF1<0y≤0，SKIPIF1<0对称轴x轴y轴焦点SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0离心率e=1准线方程SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0焦半径SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0知识点诠释：(1)与椭圆、双曲线不同，抛物线只有一个焦点、一个顶点、一条对称轴，一条准线；(2)标准方程中的参数p的几何意义是指焦点到准线的距离；p＞0恰恰说明定义中的焦点F不在准线SKIPIF1<0上这一隐含条件；参数p的几何意义在解题时常常用到，特别是具体的标准方程中应找到相当于p的值，才易于确定焦点坐标和准线方程.【典例例题】题型一：抛物线的定义例1．若P为抛物线y2=2px(p>0)上任意一点，F为抛物线的焦点，则以|PF|为直径的圆与y轴的位置关系为()A．相交 B．相离C．相切 D．不确定例2．若抛物线SKIPIF1<0上一点SKIPIF1<0到SKIPIF1<0轴的距离为SKIPIF1<0，则点SKIPIF1<0到该抛物线焦点的距离为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例3．已知抛物线SKIPIF1<0的焦点为F，SKIPIF1<0是C上一点，SKIPIF1<0，则SKIPIF1<0(

)A．1 B．2 C．3 D．4例4．抛物线的方程为SKIPIF1<0，抛物线上一点P的横坐标为SKIPIF1<0，则点P到抛物线的焦点的距离为(

)A．2 B．3 C．4 D．5例5．已知SKIPIF1<0为抛物线SKIPIF1<0：SKIPIF1<0的焦点，纵坐标为5的点SKIPIF1<0在C上，SKIPIF1<0，则SKIPIF1<0(

)A．2 B．3 C．5 D．6题型二：求抛物线的标准方程例6．点SKIPIF1<0到抛物线SKIPIF1<0的准线的距离为6，那么抛物线的标准方程是(

)A．SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0例7．准线方程为SKIPIF1<0的抛物线的标准方程是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例8．准线方程为SKIPIF1<0的抛物线的标准方程是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例9．经过点SKIPIF1<0的抛物线的标准方程是(

)A．SKIPIF1<0或SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0例10．若抛物线SKIPIF1<0上一点SKIPIF1<0到其准线的距离为3，则抛物线的标准方程为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例11．已知抛物线的准线方程为SKIPIF1<0，则该拋物线的标准方程为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例12．已知抛物线的焦点为SKIPIF1<0，则抛物线的标准方程为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0题型三：抛物线的综合问题例13．(多选题)已知抛物线SKIPIF1<0的焦点为SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是抛物线上两点，则下列结论正确的(

)A．点SKIPIF1<0的坐标为SKIPIF1<0B．若直线SKIPIF1<0经过焦点SKIPIF1<0，则SKIPIF1<0C．若SKIPIF1<0，则线段SKIPIF1<0的中点SKIPIF1<0到SKIPIF1<0轴的距离为SKIPIF1<0D．若直线SKIPIF1<0经过焦点SKIPIF1<0且满足SKIPIF1<0，则直线SKIPIF1<0的倾斜角为SKIPIF1<0例14．(多选题)斜率为1的直线l经过抛物线SKIPIF1<0的焦点F，且与抛物线相交于SKIPIF1<0两点则下列结论正确的有(

)A．SKIPIF1<0 B．抛物线的准线方程为SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例15．(多选题)已知点SKIPIF1<0，点SKIPIF1<0，点SKIPIF1<0在抛物线SKIPIF1<0上，则(

)A．当SKIPIF1<0时，SKIPIF1<0最小值为1 B．当SKIPIF1<0吋，SKIPIF1<0的最小值为4C．当SKIPIF1<0时，SKIPIF1<0的最小值为3 D．当SKIPIF1<0吋，SKIPIF1<0的最大值为2例16．(多选题)已知抛物线SKIPIF1<0的焦点SKIPIF1<0到准线的距离为SKIPIF1<0，直线SKIPIF1<0过点SKIPIF1<0且与抛物线交于A、B两点，若SKIPIF1<0是线段AB的中点，则(

)A．SKIPIF1<0 B．SKIPIF1<0 C．直线SKIPIF1<0的方程为SKIPIF1<0 D．SKIPIF1<0例17．(多选题)已知直线SKIPIF1<0过抛物线SKIPIF1<0的焦点F，且与抛物线C交于A，B两点，过A，B两点分别作抛物线准线的垂线，垂足分别为M，N，则下列说法正确的是(

)A．抛物线的方程为SKIPIF1<0 B．线段SKIPIF1<0的中点到y轴的距离为SKIPIF1<0C．线段SKIPIF1<0的长度为SKIPIF1<0 D．SKIPIF1<0题型四：轨迹方程例18．若动点SKIPIF1<0到点SKIPIF1<0的距离等于它到直线SKIPIF1<0的距离，则SKIPIF1<0点的轨迹方程是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例19．已知圆的方程为SKIPIF1<0，若抛物线过点A(﹣1，0)，B(1，0)，且以圆的切线为准线，则抛物线的焦点轨迹方程为(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例20．在平面直角坐标系xOy中，动点SKIPIF1<0到直线SKIPIF1<0的距离比它到定点SKIPIF1<0的距离小1，则P的轨迹方程为(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例21．已知动圆M与直线y=2相切，且与定圆SKIPIF1<0外切，则动圆圆心M的轨迹方程为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例22．已知在平面直角坐标系中有一定点SKIPIF1<0，动点SKIPIF1<0到y轴的距离为d，且SKIPIF1<0，则动点P的轨迹方程为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例23．与圆SKIPIF1<0：SKIPIF1<0外切，又与SKIPIF1<0轴相切的圆的圆心的轨迹方程是(

)A．SKIPIF1<0 B．SKIPIF1<0(SKIPIF1<0)和SKIPIF1<0(SKIPIF1<0)C．SKIPIF1<0(SKIPIF1<0) D．SKIPIF1<0(SKIPIF1<0)和SKIPIF1<0(SKIPIF1<0)例24．已知抛物线C：y2＝8x的焦点为F，点P是抛物线C上一动点，则线段FP的中点Q的轨迹方程是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例25．已知动圆⊙SKIPIF1<0经过定点SKIPIF1<0，且和直线SKIPIF1<0相切，则点SKIPIF1<0的轨迹方程为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0题型五：抛物线的几何性质例26．抛物线SKIPIF1<0上一点到准线和抛物线的对称轴距离分别为10和6，则该点的横坐标是__________．例27．一个正三角形的三个顶点都在抛物线SKIPIF1<0上，其中一个顶点在原点，则这个三角形的面积为__________．例28．已知抛物线SKIPIF1<0的焦点为SKIPIF1<0，过SKIPIF1<0且倾斜角为SKIPIF1<0的直线交SKIPIF1<0于SKIPIF1<0两点，线段SKIPIF1<0中点的纵坐标为SKIPIF1<0，则SKIPIF1<0__________.例29．抛物线SKIPIF1<0在第一象限上一点SKIPIF1<0，满足SKIPIF1<0，SKIPIF1<0为该抛物线的焦点，则直线SKIPIF1<0的斜率为______.例30．如图是一座抛物线型拱桥，拱桥是抛物线的一部分且以抛物线的轴为对称轴，当水面在l时，拱顶离水面2米，水面宽4米．当水位下降，水面宽为6米时，拱顶到水面的距离为______米．例31．过点SKIPIF1<0的直线SKIPIF1<0与抛物线SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0两点，点SKIPIF1<0在SKIPIF1<0轴上方，若SKIPIF1<0，则直线SKIPIF1<0的斜率SKIPIF1<0___________.例32．已知抛物线SKIPIF1<0：SKIPIF1<0经过点SKIPIF1<0，若点SKIPIF1<0到抛物线SKIPIF1<0的焦点的距离为4，则SKIPIF1<0______题型六：抛物线中的范围与最值问题例33．已知点SKIPIF1<0在抛物线SKIPIF1<0上，则点SKIPIF1<0到直线SKIPIF1<0的距离的最小值是__________．例34．已知点SKIPIF1<0为拋物线SKIPIF1<0上的动点，点SKIPIF1<0为圆SKIPIF1<0上的动点，则点SKIPIF1<0到SKIPIF1<0轴的距离与点SKIPIF1<0到点SKIPIF1<0的距离之和最小值为__________.例35．已知M为抛物线SKIPIF1<0上的动点，F为抛物线的焦点，点SKIPIF1<0，则SKIPIF1<0的最小值为__________．例36．已知抛物线SKIPIF1<0的焦点为F，定点SKIPIF1<0，点P是抛物线上一个动点，则SKIPIF1<0的最小值为______________.例37．已知点P到直线SKIPIF1<0与到点SKIPIF1<0的距离相等，点Q在圆SKIPIF1<0上，则SKIPIF1<0的最小值为_____.例38．已知抛物线SKIPIF1<0：SKIPIF1<0，SKIPIF1<0，SKIPIF1<0为SKIPIF1<0上一点，则SKIPIF1<0取最小值时点SKIPIF1<0的坐标为________．例39．SKIPIF1<0是抛物线SKIPIF1<0的焦点，P是抛物线上任一点，A(3，1)是定点，则SKIPIF1<0的最小值是___________．例40．若点SKIPIF1<0的坐标为SKIPIF1<0，F为抛物线SKIPIF1<0的焦点，点SKIPIF1<0在抛物线上移动，为使SKIPIF1<0最小，点SKIPIF1<0的坐标应为__________．例41．已知点SKIPIF1<0在抛物线SKIPIF1<0上，那么点SKIPIF1<0到点SKIPIF1<0的距离与点SKIPIF1<0到抛物线焦点距离之和取得最小值时，点SKIPIF1<0的坐标为_________例42．已知点P为抛物线C：SKIPIF1<0上的动点，直线l：SKIPIF1<0，点SKIPIF1<0为圆M：SKIPIF1<0上的动点，设点P到直线l的距离为d，则SKIPIF1<0的最小值为________．题型七：焦半径问题例43．已知抛物线SKIPIF1<0的焦点为F，SKIPIF1<0是抛物线C上一点，若SKIPIF1<0，则SKIPIF1<0________．例44．若抛物线SKIPIF1<0的焦点为SKIPIF1<0，点SKIPIF1<0在此抛物线上且横坐标为SKIPIF1<0，则SKIPIF1<0________.例45．已知抛物线SKIPIF1<0的焦点为F，直线SKIPIF1<0与抛物线交于点M，且SKIPIF1<0，则SKIPIF1<0_______.例46．过抛物线SKIPIF1<0的焦点F作直线l交抛物线C于SKIPIF1<0两点，若SKIPIF1<0，则SKIPIF1<0__________．例47．过抛物线M：SKIPIF1<0焦点的直线交抛物线M于A，B两点，若线段AB的中点P到M的准线的距离等于9，则SKIPIF1<0__________．例48．若抛物线SKIPIF1<0上一点P到焦点的距离为4，则点P到原点的距离为______．例49．抛物线SKIPIF1<0的焦点为SKIPIF1<0，过原点SKIPIF1<0的直线SKIPIF1<0交SKIPIF1<0于另一点SKIPIF1<0，若SKIPIF1<0，则SKIPIF1<0______.例50．已知抛物线SKIPIF1<0的焦点为SKIPIF1<0，点SKIPIF1<0，SKIPIF1<0在SKIPIF1<0上，SKIPIF1<0，则线段SKIPIF1<0的中点到准线的距离为___________.例51．已知抛物线SKIPIF1<0的焦点为SKIPIF1<0，准线为SKIPIF1<0，SKIPIF1<0是SKIPIF1<0上一点，SKIPIF1<0交SKIPIF1<0于SKIPIF1<0两点，且满足SKIPIF1<0，则SKIPIF1<0_____________________.例52．已知抛物线SKIPIF1<0的焦点为F，点M在抛物线上，MN垂直x轴于点N，若SKIPIF1<0，则SKIPIF1<0的面积为_________.例53．抛物线有如下光学性质：由焦点射出的光线经抛物线反射后反射光线平行于抛物线的对称轴；反之，平行于抛物线对称轴入射光线经抛物线反射后反射光线必经过抛物线的焦点．已知抛物线SKIPIF1<0的焦点为SKIPIF1<0，一平行于SKIPIF1<0轴的光线从点SKIPIF1<0射出，经过抛物线上的点SKIPIF1<0反射后，再经抛物线上的另一点SKIPIF1<0射出，则SKIPIF1<0______．例54．已知点SKIPIF1<0为抛物线SKIPIF1<0的焦点，点SKIPIF1<0在抛物线上，SKIPIF1<0为坐标原点，若SKIPIF1<0的面积为2，则SKIPIF1<0到直线SKIPIF1<0的距离为______.例55．已知抛物线SKIPIF1<0的焦点为F，点SKIPIF1<0在点F的右边，若C上的点Q满足SKIPIF1<0，则SKIPIF1<0_____________．例56．设O为坐标原点，抛物线SKIPIF1<0的焦点为F，P为抛物线上一点，若SKIPIF1<0，则SKIPIF1<0的面积为____________．【过关测试】一、单选题1．抛物线SKIPIF1<0的焦点关于直线SKIPIF1<0的对称点的坐标是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．过抛物线SKIPIF1<0的焦点SKIPIF1<0作直线，交抛物线于SKIPIF1<0，SKIPIF1<0两点，若SKIPIF1<0，则SKIPIF1<0(

)A．1 B．2 C．3 D．4

3．已知SKIPIF1<0是抛物线SKIPIF1<0：SKIPIF1<0的焦点，点SKIPIF1<0在SKIPIF1<0上且SKIPIF1<0，则SKIPIF1<0的坐标为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知抛物线SKIPIF1<0，直线SKIPIF1<0交该抛物线于SKIPIF1<0两点．若线段SKIPIF1<0的中点坐标为SKIPIF1<0，则直线SKIPIF1<0斜率为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．已知AB是经过抛物线SKIPIF1<0的焦点的弦，若点A、B的横坐标分别为1和SKIPIF1<0，则该抛物线的准线方程为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0

6．已知直线SKIPIF1<0，抛物线SKIPIF1<0，l与SKIPIF1<0有一个公共点的直线有(

)A．1条B．2条C．3条D．1条、2条或3条7．抛物线的顶点在原点，对称轴是x轴，抛物线上的点SKIPIF1<0到焦点的距离是6，则抛物线的方程为(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<08．过抛物线SKIPIF1<0的焦点F作倾斜角为SKIPIF1<0的弦AB，则SKIPIF1<0的值为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多选题9．设抛物线SKIPIF1<0的焦点为SKIPIF1<0，点SKIPIF1<0为SKIPIF1<0上一点，若SKIPIF1<0，则直线SKIPIF1<0的倾斜角可能是()A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<010．对抛物线SKIPIF1<0，下列描述正确的是()A．开口向上，焦点为SKIPIF1<0B．开口向右，准线方程为-SKIPIF1<0C．开口向右，焦点为SKIPIF1<0D．开口向上，准线方程为SKIPIF1<011．已知抛物线SKIPIF1<0：SKIPIF1<0的焦点为SKIPIF1<0，SKIPIF1<0为SKIPIF1<0上一点，且SKIPIF1<0，直线SKIPIF1<0交SKIPIF1<0于另一点SKIPIF1<0，记坐标原点为SKIPIF1<0，则(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<012．已知抛物线SKIPIF1<0的焦点为F，点P在准线上，过点F作PF的垂线且与抛物线交于A，B两点，则(

)A．SKIPIF1<0最小值为2 B．若SKIPIF1<0，则SKIPIF1<0C．若SKIPIF1<0，则SKIPIF1<0 D．若点P不在x轴上，则SKIPIF1<0三、填空题13．已知SKIPIF1<0是抛物线SKIPIF1<0的焦点，点SKIPIF1<0在抛物线SKIPIF1<0上，则SKIPIF1<0__________．14．抛物线SKIPIF1<0上的点SKIPIF1<0到焦点SKIPIF1<0的距离为SKIPIF1<0，则点SKIPIF1<0的纵坐标为________.15．抛物线SKIPIF1<0的焦点为F，过点F作斜率为SKIPIF1<0的直线l与抛物线在y轴右侧的部分相交