Re: [微分] 2題求救...
※ 引述《victor7935 (victor)》之銘言：
: 1.
: Show that
: the hyperbola x^2-y^2=5
grad A: 2x i -2y j
: and
: the ellipse 4x^2+9y^2=72
: are orthogonal.
grad B: 8x i + 18y j
二方程式聯立
13x^2 = 117
=> x^2 = 9
grad A dot grad B = 16x^2 - 36y^2
= 16x^2 - 36(x^2-5)
= -20x^2 + 180
= 0
所以在交會點處orthogonal
: 2.
: Show that the sum of the x- and y-intercepts of
: any line tangent to the graph of
: (1/2) (1/2) (1/2)
: x + y = c
: is constant and equal to c.
x, y ≧ 0
graph的grad 1/2√x i + 1/2√y j
x/a + y/b = 1 a , b > 0
(a , -b) dot (1/2√x , 1/2√y) = 0
a^2 b^2
---- = -----
x y
令x = a^2k
y = b^2k
k > 0
ak + bk = 1 => k = 1 / (a + b)
a √k + b √k = √c = √(a + b)
=> a + b = c const
--
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