过F(0,1)作直线L与抛物线y=1/4(x^2)交于两点A,B,圆C:圆心在(0,-1)半径为1的圆.过A,B分别作C的切线BD,AE,试求(|AE|^2+|BD|^2)/|AB|的取值范围.
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![过F(0,1)作直线L与抛物线y=1/4(x^2)交于两点A,B,圆C:圆心在(0,-1)半径为1的圆.过A,B分别作C的切线BD,AE,试求(|AE|^2+|BD|^2)/|AB|的取值范围.](/uploads/image/z/6992382-30-2.jpg?t=%E8%BF%87F%280%2C1%29%E4%BD%9C%E7%9B%B4%E7%BA%BFL%E4%B8%8E%E6%8A%9B%E7%89%A9%E7%BA%BFy%3D1%2F4%28x%5E2%29%E4%BA%A4%E4%BA%8E%E4%B8%A4%E7%82%B9A%2CB%2C%E5%9C%86C%EF%BC%9A%E5%9C%86%E5%BF%83%E5%9C%A8%280%2C-1%29%E5%8D%8A%E5%BE%84%E4%B8%BA1%E7%9A%84%E5%9C%86.%E8%BF%87A%2CB%E5%88%86%E5%88%AB%E4%BD%9CC%E7%9A%84%E5%88%87%E7%BA%BFBD%2CAE%2C%E8%AF%95%E6%B1%82%28%7CAE%7C%5E2%2B%7CBD%7C%5E2%29%2F%7CAB%7C%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4.)
过F(0,1)作直线L与抛物线y=1/4(x^2)交于两点A,B,圆C:圆心在(0,-1)半径为1的圆.过A,B分别作C的切线BD,AE,试求(|AE|^2+|BD|^2)/|AB|的取值范围.
过F(0,1)作直线L与抛物线y=1/4(x^2)交于两点A,B,
圆C:圆心在(0,-1)半径为1的圆.
过A,B分别作C的切线BD,AE,试求(|AE|^2+|BD|^2)/|AB|的取值范围.
过F(0,1)作直线L与抛物线y=1/4(x^2)交于两点A,B,圆C:圆心在(0,-1)半径为1的圆.过A,B分别作C的切线BD,AE,试求(|AE|^2+|BD|^2)/|AB|的取值范围.
连结CA,CE CB,CD,AE²=AC²-EC²=AC²-1,同样,BD²=BC²-1,这样所求式子简化为
(AC²+BC²-2)/AB① F恰好为抛物线的焦点,C在抛物线准线上,设A(XA,YA)B(XB,YB),则AC²=XA²+(YA+1)² BC²=XB²+(YB+1)² AB=AF+FB=YA+1+YB+1=YA+YB+2
又 XA²=4YA,XB²=4YB,带入①得
(YA²+6YA+YB²+6YB)/(YA+YB+2)②
联立抛物线和直线y=kx+1,消去x,(具体过程就是y-1=kx两边平方,把x²=4y带入右边),得到y²-(4k²+2)y+1=0 △>0 恒成立,由于过抛物线内一点,有要求有两个交点,k存在.
YA+YB=4k²+2 YA*YB=1 YA²+YB²=(YA+YB)²-2YA*YB=(4k²+2)²-2
带入②,以4k²+4为整体,凑配,分离常数得(4k²+4)-10/(4k²+4)+2
该函数单调递增.(由于-10/(4k²+4)单调递增)
所以,最小值是k=0时,为6-5/2=7/2,没有最大值,范围是【7/2,+无穷)