过抛物线y²=2x的焦点F,倾斜角为π/4的直线l交抛物线于点A、B(xA>xB),则丨AF丨/丨BF丨的值_______答案:3+2根号2,
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![过抛物线y²=2x的焦点F,倾斜角为π/4的直线l交抛物线于点A、B(xA>xB),则丨AF丨/丨BF丨的值_______答案:3+2根号2,](/uploads/image/z/12884200-16-0.jpg?t=%E8%BF%87%E6%8A%9B%E7%89%A9%E7%BA%BFy%26%23178%3B%3D2x%E7%9A%84%E7%84%A6%E7%82%B9F%2C%E5%80%BE%E6%96%9C%E8%A7%92%E4%B8%BA%CF%80%2F4%E7%9A%84%E7%9B%B4%E7%BA%BFl%E4%BA%A4%E6%8A%9B%E7%89%A9%E7%BA%BF%E4%BA%8E%E7%82%B9A%E3%80%81B%28xA%3ExB%29%2C%E5%88%99%E4%B8%A8AF%E4%B8%A8%2F%E4%B8%A8BF%E4%B8%A8%E7%9A%84%E5%80%BC_______%E7%AD%94%E6%A1%88%EF%BC%9A3%2B2%E6%A0%B9%E5%8F%B72%2C)
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过抛物线y²=2x的焦点F,倾斜角为π/4的直线l交抛物线于点A、B(xA>xB),则丨AF丨/丨BF丨的值_______答案:3+2根号2,
过抛物线y²=2x的焦点F,倾斜角为π/4的直线l交抛物线于点A、B(xA>xB),
则丨AF丨/丨BF丨的值_______答案:3+2根号2,
过抛物线y²=2x的焦点F,倾斜角为π/4的直线l交抛物线于点A、B(xA>xB),则丨AF丨/丨BF丨的值_______答案:3+2根号2,
解
抛物线y²=2x.焦点F(1/2,0)
可设直线L:y=x-(1/2).
与抛物线联立,整理可得:
x²-3x+(1/4)=0
解得:x=(3±2√2)/2
由题设可得:
xA=(3+2√2)/2,xB=(3-2√2)/2
由抛物线定义可知:
|AF|=(xA)+(1/2)
|BF|=(xB)+(1/2)
|AB|=|AF|+|BF|=(xA+xB)+1
∴|AF|/|AB|=(2|AF|)/(2|AB|)=(4+2√2)/8=(2+√2)/4
用极坐标方程:
|AF|=1/[1-cos(π/4)]=2/(2-√2),
|BF|=1/[1-cos(π+π/4)]=2/(2+√2).
∴|AF|/|BF|=(2+√2)/(2-√2)=3+2√2.