2024年数学高考大一轮复习第十三章 §13.2　参数方程

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§13.2　参数方程考试要求　1.了解参数方程，了解参数的意义.2.能选择适当的参数写出直线、圆和椭圆的参数方程．知识梳理1．参数方程和普通方程的互化(1)一般地，在平面直角坐标系中，如果曲线上任意一点的坐标x，y都是某个变数t的函数并且对于t的每一个允许值，由方程组所确定的点M(x，y)都在这条曲线上，那么此方程就叫做这条曲线的参数方程．(2)曲线的参数方程和普通方程是曲线方程的不同形式．一般地，可以通过________________而从参数方程得到普通方程．2．常见曲线的参数方程和普通方程点的轨迹普通方程参数方程直线y－y0＝tan α·(x－x0) 圆 (θ为参数)椭圆＋＝1(a>b>0) 双曲线－＝1(a>0，b>0)(φ为参数)抛物线y2＝2px(p>0)(t为参数) 思考辨析判断下列结论是否正确(请在括号中打“√”或“×”)(1)参数方程中的x，y都是参数t的函数．(　　)(2)方程(θ为参数)表示以点(0,1)为圆心，以2为半径的圆．(　　)(3)已知椭圆的参数方程(t为参数)，点M在椭圆上，对应参数t＝，点O为原点，则直线OM的斜率为.(　　)(4)参数方程(θ为参数且θ∈)表示的曲线为椭圆．(　　)教材改编题1．参数方程 (t为参数) 的图象是(　　)A．离散的点   B．抛物线C．圆   D．直线2．参数方程 (θ为参数)化为普通方程为(　　)A．x2＋＝1   B．x2＋＝1C．y2＋＝1   D．y2＋＝13．已知直线l的参数方程是(t为参数)，若l与圆x2＋y2－4x＋3＝0交于A，B两点，且|AB|＝，则直线l的斜率为________．题型一　参数方程与普通方程的互化例1 已知曲线C1，C2的参数方程为C1： (θ为参数)，C2：(t为参数)．(1)将C1，C2的参数方程化为普通方程；________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)若点P是曲线C1上的动点，求点P到C2的距离的最小值．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思维升华 消去方程中的参数一般有三种方法(1)利用解方程的技巧求出参数的表达式，然后代入消去参数．(2)利用三角恒等式消去参数．(3)根据参数方程本身的结构特征，灵活地选用一些方法从整体上消去参数．跟踪训练1 (2022·全国甲卷)在直角坐标系xOy中，曲线C1的参数方程为(t为参数)，曲线C2的参数方程为(s为参数)．(1)写出C1的普通方程；(2)以坐标原点为极点，x轴正半轴为极轴建立极坐标系，曲线C3的极坐标方程为2cos θ－sin θ＝0，求C3与C1交点的直角坐标，及C3与C2交点的直角坐标．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________题型二　参数方程的应用例2　在平面直角坐标系xOy中，曲线C的参数方程为 (λ为参数)．(1)求曲线C的普通方程；(2)已知点M(2,0)，直线l的参数方程为(t为参数)，且直线l与曲线C交于A，B两点，求＋的值．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思维升华 (1)解决直线与曲线的参数方程的应用问题时，一般是先化为普通方程，再根据直线与曲线的位置关系来解决．(2)对于形如(t为参数)的方程，当a2＋b2≠1时，应先化为标准形式后才能利用t的几何意义解题．跟踪训练2 (2022·萍乡模拟)在直角坐标系xOy中，曲线C的参数方程为C：(t为参数)，以直角坐标的原点O为极点，x轴的正半轴为极轴建立极坐标系．(1)求曲线C的极坐标方程；(2)若A，B是曲线C上的两点，且·＝0，求||的最小值．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________题型三　极坐标方程和参数方程的综合应用例3 (2022·全国乙卷)在直角坐标系xOy中，曲线C的参数方程为(t为参数)．以坐标原点为极点，x轴正半轴为极轴建立极坐标系，已知直线l的极坐标方程为ρsin＋m＝0.(1)写出l的直角坐标方程；(2)若l与C有公共点，求m的取值范围．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思维升华 解决参数方程和极坐标的综合问题的方法涉及参数方程和极坐标方程的综合题，求解的一般方法是分别化为普通方程和直角坐标方程后求解．当然，还要结合题目本身特点，确定选择何种方程．跟踪训练3 在平面直角坐标系xOy中，曲线C2的参数方程是 (α为参数)，在极坐标系(与直角坐标系xOy取相同的长度单位，且以原点为极点，以x轴正半轴为极轴)中，曲线C1的极坐标方程是ρcos θ－3＝0，点P是曲线C2上的动点．(1)求点P到曲线C1的距离的最大值；________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)若曲线C3：θ＝交曲线C2于A，B两点，求△ABC2的面积．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________