高中数学题,椭圆从椭圆x2/x2+y2/b2=1(a>b>0)上一点P向X轴作垂线,垂足恰为椭圆的右焦点F2,A是椭圆的

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  • 直线l 过点F2(c,0)及点Q(a,b),

    ∴l:x=(a-c)y/b+c,①

    代入椭圆方程:b^2x^2+a^2y^2=a^2b^2得

    (a-c)^2y^2+2bc(a-c)y+b^2c^2+a^2y^2=a^2b^2,

    整理得(2a^2-2ac+c^2)y^2+2bc(a-c)y-b^4=0,

    设M(x1,y1),N(x2,y2),l上的点R((a-c)m+c,bm),则

    y1+y2=-2bc(a-c)/(2a^2-2ac+c^2),y1y2=-b^4/(2a^2-2ac+c^2)

    向量MR·NR= -2,得

    ((a-c)m+c-x1,bm-y1)*((a-c)m+c-x2,bm-y2)

    =[(a-c)m+c-x1][(a-c)m+c-x2]+(bm-y1)(bm-y2)

    =(a-c)^2(m-y1/b)(m-y2/b)+(bm-y1)(bm-y2)(由①)

    =[(a-c)^2/b^2+1][b^2m^2-bm(y1+y2)+y1y2]

    =[(a-c)^2/b^2+1][b^2m^2(2a^2-2ac+c^2)+2b^2cm(a-c)-b^4]/(2a^2-2ac+c^2)=-2,

    ∴m^2(2a^2-2ac+c^2)+2cm(a-c)-b^2=-2(2a^2-2ac+c^2)/[(a-c)^2+b^2],②

    OR^2=[(a-c)m+c]^2+b^2m^2

    =[(a-c)^2+b^2]m^2+2c(a-c)m+c^2,③

    在约束条件②之下求③的最小值,繁!