已知实数x,y满足方程x^2+y^2-4x-2y+4=0. （1）求x+y的最小值和最大值。 （2）求y/x的取值范围。
\u5df2\u77e5\u5b9e\u6570x\uff0cy\u6ee1\u8db3\u65b9\u7a0bx^2+y^2-4x=0\uff081\uff09\u6c42y-2\uff0fx+1\u7684\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c\u5706\u65b9\u7a0b\u5373\uff1a(x-2)²+y²\uff1d(\u221a3)².
(1)\u8bbe(y+2)/(x+1)\uff1dt\u2192tx-y+t-2\uff1d0.
\u6b64\u76f4\u7ebf\u7cfb\u4e0e\u5706\u5fc3(2,0)\u7684\u8ddd\u79bb\u4e0d\u5927\u4e8e\u534a\u5f84\u221a3\uff0c
\u2234|t\u00b72-0+t-2|/\u221a(t²+1)\u2264\u221a3
\u21926t²-12t+1\u22640
\u2192(6-\u221a30)/6\u2264t\u2264(6+\u221a30)/6.
\u6240\u6c42\u6700\u5927\u503c\uff1a(6+\u221a30)/6;
\u6240\u6c42\u6700\u5c0f\u503c\uff1a(6-\u221a30)/6.
(2)\u8bbex-2y\uff1d\u03bb,\u5b83\u4e0e\u5706\u5fc3(2,0)\u8ddd\u79bb\u4e0d\u5927\u4e8e\u534a\u5f84\u221a3\uff0c
\u2234|2-2\u00b70-\u03bb|/\u221a[1²+(-2)²]\u2264\u221a3
\u21922-\u221a15\u2264\u03bb\u22642+\u221a15.
\u6240\u6c42\u6700\u5c0f\u503c\u4e3a\uff1a2-\u221a15\uff1b
\u6240\u6c42\u6700\u5927\u503c\u4e3a\uff1a2+\u221a15.
(3)\u5706\u4e0a\u70b9P\u53ef\u8bbe\u4e3a(2+\u221a3cos\u03b8,\u221a3sin\u03b8).
\u70b9P\u4e0e3x+4y+12\uff1d0\u7684\u8ddd\u79bb\u4e3a\uff1a
d\uff1d|3(2+\u221a3cos\u03b8)+4\u221a3sin\u03b8+12|/\u221a(3²+4²)
\uff1d|18+5\u221a3sin(\u03b8+\u03c6)|/5
(\u5176\u4e2dtan\u03c6\uff1d3/4)
sin(\u03b8+\u03c6)\uff1d1\u65f6\uff0c\u6240\u6c42\u6700\u5927\u503c\u4e3a\uff1a(18+5\u221a3)/5;
sin(\u03b8+\u03c6)\uff1d-1\u65f6,\u6240\u6c42\u6700\u5c0f\u503c\u4e3a\uff1a(18-5\u221a3)/5\u3002
\u539f\u65b9\u7a0b\u53ef\u5316\u4e3a\uff1a
(x-2)²+y²=3
\u8fd9\u662f\u5706\u5fc3\u5728\uff082,0\uff09\u534a\u5f84\u7b49\u4e8e\u221a3\u7684\u5706\uff0c\u6ee1\u8db3\u8be5\u65b9\u7a0b\u7684\u70b9P(x,y)\u5728\u5706\u4e0a,\u5e76\u4e14y/x\u4e3a\u76f4\u7ebfOP\u7684\u659c\u7387\u3002
\u663e\u7136\uff0c\u5f53OP\u4e0e\u5706\u76f8\u5207\uff0c\u5e76\u4e14\u4f4d\u4e8e\u7b2c\u4e00\u8c61\u9650\u65f6\uff0c\u5176\u659c\u7387\u6700\u5927\u3002
\u4ee4OP\u7684\u65b9\u7a0b\u4e3a y=kx\uff0c\u4ee3\u5165\u539f\u65b9\u7a0b\u5f97
(1+k²)x²-4x+1=0
\u4ee4\u5224\u522b\u5f0f \u25b3=16-4(1+k²)=0
\u89e3\u51fak\u5f97\uff1ak=\u00b1\u221a3
\u6700\u540e\u5f97\u5230\uff1ay/x\u7684\u6700\u5927\u503c\u4e3a\u221a3,\u6700\u5c0f\u503c\u4e3a-\u221a3
\u6ce8\u610f\u5230\u5df2\u77e5\u65b9\u7a0b\u7684\u56fe\u5f62\u662f\u5706\uff0c\u7528\u6570\u5f62\u7ed3\u5408\u7684\u601d\u60f3\u5c31\u53ef\u4ee5\u5f88\u5feb\u627e\u51fa\u89e3\u9898\u65b9\u5411\u3002
所以x^2+y^2-4x-2y+4+1=1
即(x-2)^2 + (y-1)^2 = 1
可令x = 2+sina, y = 1+cosa， 0<=a<2PI
(1)
x+y = 3 + sina + cosa = 3 + (根号2)*sin(PI/4 + a)
所以3-根号2<=x+y<= 3+根号2
最小值是3-根号2，最大值是3+根号2
(2)
y/x = (1+cosa) / (2+sina)
= 2cos(a/2)^2 / (2 + 2sin(a/2)cos(a/2))
= 1 / (sec(a/2)^2 + tan(a/2))
= 1 / (tan(a/2)^2 + tan(a/2) + 1)
= 1 / [(tan(a/2) + 1/2)^2 + 3/4]
因为(tan(a/2) + 1/2)^2 >= 0
所以0 < y/x <= 1 / (3/4) = 4/3
即取值范围为 0 < y/x <= 4/3
1）(x-2)^2+(y-1)^2=1 设x=2+cosθ y=1+sinθ x+y=3+sinθ+cosθ=3+根号2sin(θ+π/4)
所以最大值为3+根号2 最小值3-根号2
2）y/x=y-0/x-0为圆上的点到原点的斜率 范围画图可以求出希望采纳
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绛旓細璁:M=X-2Y y=(x-m)/2 X^2-2X-4Y=5 x^2-2x-4*(x-m)/2=5 x^2-2x-2x+2m-5=0 x^2-4x+2m-5=0 鍒ゅ埆寮廱^2-4ac=16-4(2m-5)=36-8m m<=36/8=9/2 鎵浠鍙栧艰寖鍥存槸m<=9/2
