已知双曲线x^2/2-y^2/2=1(x大于等于根号2)上有不同的两点A,B，o是坐标原点，求向量OA.OB数量积的最小值 已知双曲线x^2/a^2-y^2/2=1(a大于根号2)
\u5df2\u77e5\u53cc\u66f2\u7ebfx^2/2-y^2/2=1(x\u5927\u4e8e\u7b49\u4e8e\u6839\u53f72)\u4e0a\u6709\u4e0d\u540c\u7684\u4e24\u70b9A,B\uff0co\u662f\u5750\u6807\u539f\u70b9\uff0c\u6c42\u5411\u91cfOA.OB\u6570\u91cf\u79ef\u7684\u6700\u5c0f\u503c\u7531x\u7684\u8303\u56f4\u77e5\u9053A\u3001B\u4e24\u70b9\u5728\u66f2\u7ebf\u53f3\u652f\u4e0a,\u66f2\u7ebf\u65b9\u7a0b\u53d8\u5f62\u4e3a\uff1ax^2=y^2+2\u5176\u4e2dx\u2265\u221a2 , \u8bbeA\u3001B\u7eb5\u5750\u6807\u5206\u522b\u662fy1\u3001y2,\u5219\u6a2a\u5750\u6807\u662f\u221a(y1^2+2)\u3001\u221a(y2^2+2),,\u6839\u636e\u6570\u91cf\u79ef\u7684\u5750\u6807\u516c\u5f0f(\u5411\u91cfa\u2299b=x1x2+y1y2)\u5f97\uff1a
\u5411\u91cfOA\u4e0eOB\u7684\u6570\u91cf\u79ef\uff1a\u221a(y1^2+2)\u00d7\u221a(y2^2+2)+y1\u00d7y2
\u5206\u6790\u6b64\u8868\u8fbe\u5f0f\uff0c\u53ef\u4ee5\u770b\u51fa \u5982\u679c\u8981\u6570\u91cf\u79ef\u6700\u5c0f\uff0c\u5219y1\u4e0ey2\u7684\u7b26\u53f7\u8981\u76f8\u53cd\uff0c\u5e76\u4e14\u5f53y1=-y2\u65f6\uff0c\u53d6\u5f97\u6700\u5c0f\u503c2\uff0c
\u6240\u4ee5\u6700\u5c0f\u503c\u662f2
A
\u6e10\u8fd1\u7ebf\u659c\u7387\u4e3a \u6839\u53f72/a
\u503e\u659c\u89d2\u662f30\u621660\u5ea6\uff0c
\u6839\u53f72/a=tan30\u6216tan60\uff0c\u6709\u4e00\u4e2a\u4e0d\u6210\u7acb
\u89e3\u5f97a=\u6839\u53f76
\u79bb\u5fc3\u7387\u662f\u4e09\u5206\u4e4b\u4e8c\u500d\u6839\u4e09
设A(x1,y1)
B(x2,y2)
x1>0,x2>0
且x1x2≥2
向量OA*向量OB
=x1x2+y1y2
≥x1x2-√(x1²-2)*√(x2²-2)
=x1x2-√[(x1x2)²-2(x1²+x2²)+4]
≥x1x2-√[(x1x2)²-4x1x2+4]
=x1x2-√(x1x2-2)²
=x1x2-|x1x2-2|
=x1x2-(x1x2-2)
=2
所以向量OA*向量OB的最小值是2
由x的范围知道A、B两点在曲线右支上,曲线方程变形为：x^2=y^2+2其中x≥√2 , 设A、B纵坐标分别是y1、y2,则横坐标是√(y1^2+2)、√(y2^2+2),,根据数量积的坐标公式(向量a⊙b=x1x2+y1y2)得：
向量OA与OB的数量积：√(y1^2+2)×√(y2^2+2)+y1×y2
分析此表达式，可以看出 如果要数量积最小，则y1与y2的符号要相反，并且当y1=-y2时，取得最小值2，
所以最小值是2
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