已知中心在原点,焦点在坐标轴上的椭圆与直线x+y=1相交于A,B两点,

3个回答

  • mx^2+ny^2=1

    x+y=1

    mx^2+n(1-x)^2=1

    (m+n)x^2-2nx+n^2-1=0

    x1+x2=2n/(m+n)

    x1x2=(n^2-1)/(m+n)

    (x1-x2)^2=(x1+x2)^2-4x1x2

    =4n^2/(m+n)^2-4(n^2-1)/(m+n)

    |AB|^2=2(x1-x2)^2=8

    Cx=(x1+x2)/2=n/(m+n) Cy=1-n/(m+n) =m/(m+n)

    Cy/Cx=m/n=√2/2

    m=√2/2n

    (x1-x2)^2=4 n^2/(m+n)^2-(n^2-1)/(m+n)=1

    1/(1+√2/2)-(n^2-1)/(1+√2/2)n=1

    n-(n^2-1)=(1+√2/2)n

    1-n^2=√2n/2

    (n+√2/4)^2=1+1/8

    n+√2/4=3/√8

    n=√2/2

    m=1/2

    x^2/2+y^2/√2=1