作者LuisSantos (^______^)
看板trans_math
标题Re: [微分]
时间Wed Feb 7 02:33:06 2007
※ 引述《king911015 (早已放弃爱上你)》之铭言：
: The shortest distance form the point (0,2) to the hyperbola X^2 -Y^2=1 is?
令 P(x,y) 为双曲线 X^2 - Y^2 = 1上的一点
则 x^2 - y^2 = 1 => x^2 = y^2 + 1
令 A (0,2)
(线段PA)^2 = (x - 0)^2 + (y - 2)^2
= x^2 + y^2 - 4y + 4
= y^2 + 1 + y^2 - 4y + 4
= 2y^2 - 4y + 5
令 g(y) = 2y^2 - 4y + 5
则 g'(y) = 4y - 4 , g"(y) = 4
令 g'(y) = 0 => 4y - 4 = 0 => y = 1
g"(1) = 4 > 0
因此当 y = 1 时 , g(y)有最小值g(1) = 2 - 4 + 5 = 3
此时 x^2 = y^2 + 1 = 1 + 1 = 2
x = √2 或 -√2
最短距离为√3
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 61.66.173.21
1F：推 kanx:都没想到有这个方法 orz 220.133.130.61 02/07 02:40