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以小求大:观察面积间的比例关系(2014湖南圆锥曲线)

2022-10-11 11:54 作者:数学老顽童  | 我要投稿

(2014湖南,21)如图,O为坐标原点,椭圆C_1%5Cfrac%7Bx%5E2%7D%7Ba%5E2%7D%2B%5Cfrac%7By%5E2%7D%7Bb%5E2%7D%3D1a%3Eb%3E0)的左、右焦点分别为F_1F_2,离心率为e_1;双曲线C_2%5Cfrac%7Bx%5E2%7D%7Ba%5E2%7D-%5Cfrac%7By%5E2%7D%7Bb%5E2%7D%3D1的左、右焦点分别为F_3F_4,离心率为e_2,已知e_1e_2%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D,且%5Cleft%7C%20F_2F_4%20%5Cright%7C%3D%5Csqrt%7B3%7D-1.

(1)求C_1C_2的方程;

(2)过F_1C_1的不垂直于y轴的弦ABMAB的中点,当直线OMC_2交于PQ两点,求四边形APBQ面积的最小值.

解:(1)由题可知:

%5Cbegin%7Bcases%7D%09%5Cfrac%7B%5Csqrt%7Ba%5E2-b%5E2%7D%7D%7Ba%7D%5Ccdot%20%5Cfrac%7B%5Csqrt%7Ba%5E2%2Bb%5E2%7D%7D%7Ba%7D%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%2C%5C%5C%09%5Csqrt%7Ba%5E2%2Bb%5E2%7D-%5Csqrt%7Ba%5E2-b%5E2%7D%3D%5Csqrt%7B3%7D-1.%5C%5C%5Cend%7Bcases%7D

解得:%5Ccolor%7Bred%7D%7Ba%3D%5Csqrt%7B2%7D%7D%5Ccolor%7Bred%7D%7Bb%3D1%7D

所以C_1C_2的方程分别为:

%5Ccolor%7Bred%7D%7B%5Cfrac%7Bx%5E2%7D%7B2%7D%2By%5E2%3D1%7D%5Ccolor%7Bred%7D%7B%5Cfrac%7Bx%5E2%7D%7B2%7D-y%5E2%3D1%7D.

如图,连接OAOB

设直线AB的方程为x%3Dmy-1,

与椭圆联立得

%5Cleft(%20m%5E2%2B2%20%5Cright)%20y%5E2-2my-1%3D0

依韦达定理:

y_1%2By_2%3D%5Cfrac%7B2m%7D%7Bm%5E2%2B2%7Dy_1y_2%3D%5Cfrac%7B-1%7D%7Bm%5E2%2B2%7D

所以

%5Ccolor%7Bred%7D%7By_M%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft(%20y_1%2By_2%20%5Cright)%20%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20%5Cfrac%7B2m%7D%7Bm%5E2%2B2%7D%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7Bm%7D%7Bm%5E2%2B2%7D%7D

所以

%5Ccolor%7Bred%7D%7Bx_M%7D%3Dmy_M-1%3Dm%5Ccdot%20%5Cfrac%7Bm%7D%7Bm%5E2%2B2%7D-1%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7B-2%7D%7Bm%5E2%2B2%7D%7D

所以

%5Ccolor%7Bred%7D%7Bk_%7BOM%7D%7D%3D%5Cfrac%7By_M-0%7D%7Bx_M-0%7D%3D%5Cfrac%7B%5Cfrac%7Bm%7D%7Bm%5E2%2B2%7D-0%7D%7B%5Cfrac%7B-2%7D%7Bm%5E2%2B2%7D-0%7D%3D%5Ccolor%7Bred%7D%7B-%5Cfrac%7Bm%7D%7B2%7D%7D

所以直线%5Ccolor%7Bred%7D%7BOM%7D的方程为%5Ccolor%7Bred%7D%7By%3D-%5Cfrac%7Bm%7D%7B2%7Dx%7D.

与双曲线联立,消去y,解得

%5Ccolor%7Bred%7D%7Bx_P%3D%5Cfrac%7B-2%7D%7B%5Csqrt%7B2-m%5E2%7D%7D%7D.

显然,m%5E2%5Cin%20%5Ccolor%7Bred%7D%7B%5Cleft%5B%200%2C2%20%5Cright)%20%7D%20.

所以

%5Ccolor%7Bred%7D%7B%5Cfrac%7B%5Cleft%7C%20PQ%20%5Cright%7C%7D%7B%5Cleft%7C%20OM%20%5Cright%7C%7D%7D%3D%5Cfrac%7B2%5Cleft%7C%20x_P%20%5Cright%7C%7D%7B%5Cleft%7C%20x_m%20%5Cright%7C%7D%3D%5Cfrac%7B2%5Cleft%7C%20%5Cfrac%7B-2%7D%7B%5Csqrt%7B2-m%5E2%7D%7D%20%5Cright%7C%7D%7B%5Cleft%7C%20%5Cfrac%7B-2%7D%7Bm%5E2%2B2%7D%20%5Cright%7C%7D%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7B2%5Cleft(2%2Bm%5E2%5Cright)%7D%7B%5Csqrt%7B2-m%5E2%7D%7D%7D

%5Cbigtriangleup%20OAB的水平宽为%5Ccolor%7Bred%7D%7B%5Cvert%20OF_1%20%5Cvert%20%3D1%7D

其铅锤高为

%5Cbegin%7Baligned%7D%0A%09%5Ccolor%7Bred%7D%7B%5Cleft%7C%20y_1-y_2%20%5Cright%7C%7D%26%3D%5Csqrt%7B%5Cleft(%20y_1%2By_2%20%5Cright)%20%5E2-4y_1y_2%7D%5C%5C%0A%09%26%3D%5Csqrt%7B%5Cleft(%20%5Cfrac%7B2m%7D%7Bm%5E2%2B2%7D%20%5Cright)%20%5E2-4%5Ccdot%20%5Cleft(%20%5Cfrac%7B-1%7D%7Bm%5E2%2B2%7D%20%5Cright)%7D%5C%5C%0A%09%26%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7B2%5Csqrt%7B2%7D%5Ccdot%20%5Csqrt%7Bm%5E2%2B1%7D%7D%7Bm%5E2%2B2%7D%7D%5C%5C%0A%5Cend%7Baligned%7D

所以

%5Cbegin%7Baligned%7D%0A%09S_%7BAPQB%7D%26%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7B%5Cleft%7C%20PQ%20%5Cright%7C%7D%7B%5Cleft%7C%20OM%20%5Cright%7C%7D%5Ccdot%20S_%7B%5Cbigtriangleup%20OAB%7D%7D%5C%5C%0A%09%26%3D%5Cfrac%7B%5Cleft%7C%20PQ%20%5Cright%7C%7D%7B%5Cleft%7C%20OM%20%5Cright%7C%7D%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20%5Cleft%7C%20OF_1%20%5Cright%7C%5Ccdot%20%5Cleft%7C%20y_1-y_2%20%5Cright%7C%5C%5C%0A%09%26%3D%5Cfrac%7B2%5Cleft(%202%2Bm%5E2%20%5Cright)%7D%7B%5Csqrt%7B2-m%5E2%7D%7D%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%5Ccdot%201%5Ccdot%20%5Cfrac%7B2%5Csqrt%7B2%7D%5Ccdot%20%5Csqrt%7Bm%5E2%2B1%7D%7D%7Bm%5E2%2B2%7D%5C%5C%0A%09%26%3D2%5Csqrt%7B2%7D%5Ccdot%20%5Csqrt%7B%5Cfrac%7Bm%5E2%2B1%7D%7B2-m%5E2%7D%7D%5C%5C%0A%09%26%3D%5Ccolor%7Bred%7D%7B2%5Csqrt%7B2%7D%5Ccdot%20%5Csqrt%7B%5Cfrac%7B3%7D%7B2-m%5E2%7D-%5Cfrac%7B1%7D%7B2%7D%7D%5Cgeqslant%202%5Csqrt%7B2%7D%7D%5C%5C%0A%5Cend%7Baligned%7D

当且仅当m%3D0时取得最小值.


以小求大:观察面积间的比例关系(2014湖南圆锥曲线)的评论 (共 条)

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