椭圆X^2/25+Y^2/9=1上不同三点A(x1,y1),B(4,9/5),C(x2,y2)与焦点F(4.0)的距离成等差数列,求证ac的垂直平分线过定点
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/27 13:54:54
![椭圆X^2/25+Y^2/9=1上不同三点A(x1,y1),B(4,9/5),C(x2,y2)与焦点F(4.0)的距离成等差数列,求证ac的垂直平分线过定点](/uploads/image/z/5453220-12-0.jpg?t=%E6%A4%AD%E5%9C%86X%5E2%2F25%2BY%5E2%2F9%3D1%E4%B8%8A%E4%B8%8D%E5%90%8C%E4%B8%89%E7%82%B9A%28x1%2Cy1%29%2CB%284%2C9%2F5%29%2CC%28x2%2Cy2%29%E4%B8%8E%E7%84%A6%E7%82%B9F%284.0%29%E7%9A%84%E8%B7%9D%E7%A6%BB%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E6%B1%82%E8%AF%81ac%E7%9A%84%E5%9E%82%E7%9B%B4%E5%B9%B3%E5%88%86%E7%BA%BF%E8%BF%87%E5%AE%9A%E7%82%B9)
椭圆X^2/25+Y^2/9=1上不同三点A(x1,y1),B(4,9/5),C(x2,y2)与焦点F(4.0)的距离成等差数列,求证ac的垂直平分线过定点
椭圆X^2/25+Y^2/9=1上不同三点A(x1,y1),B(4,9/5),C(x2,y2)与焦点F(4.0)的距离成等差数列,求证ac的垂直平分线过定点
椭圆X^2/25+Y^2/9=1上不同三点A(x1,y1),B(4,9/5),C(x2,y2)与焦点F(4.0)的距离成等差数列,求证ac的垂直平分线过定点
由已知三距离成等差数列 |AF|-|BF|=|BF|-|CF| ,得:
2*|BF|=|AF|+|CF| → 2*9/5=[ (x1-4)^2 + y1^2 ]^(1/2) + [ (x2-4)^2 + y2^2 ]^(1/2)
因为(x1,y1),(x2,y2)在圆上,x1^2/25+y1^2/9=1,x2^2/25+y2^2/9=1,带入上面方程得:
2*9/5 = [ (25-4*x1)^2/25 ]^(1/2) + [ (25-4*x2)^2/25 ]^(1/2)
2*9/5 = [ 50 - 4(x1+x2) ]/5
∴ x1+x2=8 ;①
又∵ y1^2-y2^2=(-9/25)*(x1^2-x2^2)=(-72/25)*(x1-x2)
∴ -(x1-x2) / (y1-y2) = (25/72)*(y1+y2) ②
AC的垂直平分线过AC中点[ (x1+x2)/2,(y1+y2)/2 ],且与AC垂直,其方程可表示为:
y = [-(x1-x2) / (y1-y2) ] * [ x-(x1+x2)/2 ] + (y1+y2)/2 根据①、②式可得:
y = (25/72)*(y1+y2) * [ x-64/25 ]
所以,AC垂直平分线必过 ( 64/25,0 ) 点.