过点P(-1,1)作直线与椭圆x^2/4+y^2/2=1交于AB两点,若线段AB的中点为P,则弦长|AB|=

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  • xA+xB=2xP=-2,

    yA+yB=2yP=2

    k(AB)=(yA-yB)/(xA-xB)

    x^2/4+y^2/2=1

    x^2+2y^2=4

    [(xA)^2+2(yA)^2]-[(xB)^2+2(yB)^2]=4-4=0

    (xA+xB)*(xA-xB)+2(yA+yB)*(yA-yB)=0

    (xA+xB)+2(yA+yB)*(yA-yB)/(xA-xB)=0

    -2+2*2k(AB)=0

    k(AB)=0.5

    直线的方程:x-2y-3=0

    x=2y+3

    x^2+2y^2=4

    (2y+3)^2+2y^2=4

    6y^2+12y+5=0

    yA+yB=-2,

    yA*yB=5/6

    (yA-yB)^2=(yA+yB)^2-4yA*yB=4-4*5/6=2/3

    (xA-xB)^2=4(yA-yB)^2

    AB^2=(xA-xB)^2+(yA-yB)^2=5(yA-yB)^2=10/3

    |AB|=√(10/3)