2024年数学高考大一轮复习第十二章 §12.2　参数方程（附答单独案解析）

§12.2　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

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＋＝1

C．y2＋＝1   D．y2＋＝1

3．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；

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(2)若点P是曲线C1上的动点，求点P到C2的距离的最小值．

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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．

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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．

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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 (2023·榆林模拟)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．

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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．

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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；

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(2)若曲线C3：θ＝交曲线C2于A，B两点，求△ABC2的面积．

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