Re: [考古] 100成大
※ 引述《ny37594133 (ny3759413)》之銘言：
: Find the points in the curve 17x^2+12xy+8y^2=100 that are closet to and
: farthest away from the origin.
: 麻煩幫幫我解這題!!
這個可以動一點手腳(1) 或者透過隱函數微分(2)
(1) 考慮 矩陣
M = [17,6;6,8] => eigvalues 5, 20
對應的 eigenvectors 1/sqrt(5)*[1,-2]^T := v_1
1/sqrt(5)*[2,1]^T := v_2
=> 透過 e_1 = sqrt(5)/5 * v_1 + 2*sqrt(5)/5 * v_2
e_2 = -2*sqrt(5)/5 * v_1 + sqrt(5)/5 * v_2
得到 5 * v_1^2 + 20 * v_2^2 = 100
令橢圓的參數式 然後下去解
(2) 假定 y= y(x)
=> s^2(x) = x^2 + y^2
=> 2s * s' = 2x + 2y * y'
-x
當 s' = 0 => y' = -----
y
-17 x - 6y
另外 y' = -----------------
6x + 8y
=> 當 s' = 0 , (2x+y)(x-2y) = 0
=> x = 2y 或 x = -y/2
帶入曲線得4點 (-2,-1), (2,1)
(2,-1), (-2,1)
10
當 s 不可微 => y = 0 => 帶入曲線得 x = +/- --------------
sqrt(17)
所以和原點距離最長和最短的距離是 10/sqrt(17) , 1/sqrt(5)
有錯請指正
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