已知圆C过定点A(0,P)(P为正数,圆心C在抛物线X=2Py上运动,若MN为圆C在X轴上截得的弦,设AM=L1,AN=

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  • 已知圆C过定点A(0,P)(P为正数,圆心C在抛物线X^2=2Py上运动,若MN为圆C在X轴上截得的弦,设AM=L1,AN=L2,〈MAN=θ (1)当C点运动时,MN是否变化,并证明(2)求式子L2/L1+L1/L2的最大值,并求取得此最大值时θ的值和此圆的方程

    设圆心(2t,2tt/p)

    (x-2t)^2+(y-2tt/p)^2=rr

    过定点A(0,P)

    4tt+pp+4tttt/pp-4tt=rr

    pp+4tttt/pp=rr

    (x-2t)^2+(y-2tt/p)^2=rr

    (x-2t)^2+4tttt/pp=rr=pp+4tttt/pp

    (x-2t)^1=ppx1=2t+px2=2t-pMN=2p定值当C点运动时MN不变化=2P

    L1L1=4tt+2pp+4tp

    L2L2=4tt+2pp-4tp

    L1L1+L2L2=MNMN+2L1L2cosθ

    s=L2/L1+L1/L2

    ss-2=(4tt+2pp+4tp)/(4tt+2pp-4tp) +(4tt+2pp-4tp)/(4tt+2pp+4tp)

    =2+8tp/(4tt+2pp-4tp) -8tp/(4tt+2pp+4tp)

    =2 +8tp(8tp)/(4tt+2pp-4tp) (4tt+2pp+4tp)

    =2+16pp *tt/(2tt+pp-2tp) (2tt+pp+2tp)

    =2+16pp/(2t +pp/t -2p)(2t +pp/t +2p)

    <=2+16pp/(2genhao2p -2p)(2genhao2p +2p)=6

    L2/L1+L1/L2的最大值=6

    当2t =pp/t时取到