標題:
simple differential equation p
發問:
一條很簡單的題目Use appropiate differential equation, to prove that :for all parallel light rays traveling in the negative y direction, a concave mirror reflects the light rays such that all the light rays pass through the origin. The shape of the mirror must be parabola. and find the equation of the shape... 顯示更多 一條很簡單的題目 Use appropiate differential equation, to prove that : for all parallel light rays traveling in the negative y direction, a concave mirror reflects the light rays such that all the light rays pass through the origin. The shape of the mirror must be parabola. and find the equation of the shape of the concave mirror 更新: -藍閃蝶，要prove 啊 而且你個ans 都錯左
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最佳解答:
如圖: 圖片參考：http://s585.photobucket.com/albums/ss296/mathmanliu/parabola.gif 或參考http://www.wretch.cc/album/show.php?i=mathmanliu&b=1&f=1740510794&p=113 曲線上點P(x, y)處, 切線與PO與水平線夾角相等, 則 切線與水平線夾角tan值= dy/dx = y' 切線與PO連線夾角tan值= (y/x - y')/(1+ y' * y/x) => y'= (y- xy')/(x+yy'), y y'^2 + 2xy' - y=0 => yy' = -x + √(x^2+y^2) -----(A) (Note: -x - √(x^2+y^2) 不合) 設 u= √(x^2+y^2) => u^2 = x^2+y^2 => uu'=x+yy' 代入(A)式 => uu'- x = -x + u , u'= 1 => u= x+ 2c, u^2= (x+2c)^2 => y^2= 4cx+ 4c^2 y^2= 4c(x+c) 為拋物線(頂點(- c, 0), 焦點(0, 0))
其他解答:
the equation of the shape of the concave mirror y = a - 1/(4a) * x2 where a is a positive real number
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