注意椭圆参数方程{x=acosθ y=bsinθ}中参数θ的意义

• 作 者: 陶德河
• 机 构: 安徽省和县一中,
• 出 处: 中学数学
• 发表日期: 1986年06期
• 关键词: 参数方程 普通方程 参数θ 可证 点坐标 极坐标方程 离心角 辐角 错解 切一

题目 A、B为椭圆b~2x~2+a~2y~2=a~2b~2(a>b>0)上的两点,O为中心,OA⊥OB;求1/OA+1/OB的南的最大值和最小值。错解化椭圆的普通方程为参数方程x=acosθ y=bsinθ (θ为参数) 设A、B两点的坐标分别(acosθ_1,bs nθ_1),(cosθ_2,bsinθ_2)。由OA⊥OB得θ_2+θ_1±π/2,则B点坐标为(±asinθ_1,bcosθ_1)。可证 1/(OA)~2+1/(OB)~2=(a~2+b~2)/a~2b~2。则有 (1/OA)+(1/OP)~2=(a~2+b~2)/(a~2b~2)+2/(OA·OB) =(a~2+b~2)/(a~2b~2)+2/(a~2b~2+(a~2-b~2)/2))~2sn~2θ_1

Topics A and B are two points on the ellipse b~2x~2+a~2y~2=a~2b~2(a>b>0). O is the center and OA is OB; find 1/OA+1/ The maximum and minimum values ​​of South of OB. The ordinary equation for misinterpreting an ellipse is the parametric equation x=acosθ y=bsinθ (θ is a parameter) Let A and B be the two coordinates (acosθ_1, bs nθ_1), (cosθ_2, bsinθ_2). If θ 2+ 2+ 2 + θ ± 1 π π/2 is obtained by OA ⊥ OB, the coordinate of B point is (±asin θ , 1, bcos θ _1 1). It can be proved that 1/(OA)~2+1/(OB)~2=(a~2+b~2)/a~2b~2. (1/OA)+(1/OP)~2=(a~2+b~2)/(a~2b~2)+2/(OA·OB)=(a~2+b~2) )/(a~2b~2)+2/(a~2b~2+(a~2-b~2)/2))~2sn~2θ_1