抛物线G：y²=4x,A、B为G上异于原点的两点,FA⊥FB,延长AF、BF交G于C、D求四边形ABCD面积的最小值

抛物线G：y²=4x,A、B为G上异于原点的两点,FA⊥FB,延长AF、BF交G于C、D求四边形ABCD面积的最小值
抛物线G：y²=4x,A、B为G上异于原点的两点,FA⊥FB,延长AF、BF交G于C、D求四边形ABCD面积的最小值

抛物线G：y²=4x,A、B为G上异于原点的两点,FA⊥FB,延长AF、BF交G于C、D求四边形ABCD面积的最小值
两直线垂直,焦点为(1,0),不妨设两直线为:y=k(x-1)与ky=1-x
分别与抛物线方程连立(因为有两个交点,所以k≠0):
y=k(x-1).(1)
y^2=4x.(2)
代入有k^2x^2-2k^2x+k^2-4x=0,k^2x^2-2(k^2+2)x+k^2=0
|x1-x2|=√Δ/|a|=4√(k^2+1)/k^2
弦长L1=√(k^2+1)|x1-x2|=4(k^2+1)/k^2
同理,再连立一次
ky=1-x.(1)
y^2=4x.(2)
代入有y^2=4-4ky,y^2+4ky-4=0
|y1-y2|=√Δ/|a|=4√(k^2+1)
弦长L2=√(k^2+1)|y1-y2|=4(k^2+1)
两条直线相互垂直,这个四边形的面积S=0.5L1*L2=0.5*4×4(k^2+1)^2/k^2
S=8[k^2+(1/k^2)+2]≥32
当且仅当k=±1时等号成立,此时取到面积最小值为32

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