椭圆x^2/a^2+y^2/b^2=1(a>b>0)的上顶点为A,左顶点为B,F为右焦点,过F作平行
A(0,b),B(-a,0),F(c,0),O(0,0)。设:C(x1,y1),D(x2,y2),E(s,t)
Kcd =(y1-y2)/(x1-x2) = Kab =b/a 。。。(1),(K:直线斜率)
C,D两点在椭圆上: x1^2/a^2+y1^2/b^2=1, x2^2/a^2+y2^2/b^2=1
==> (x1-x1)(x1+x2)/a^2 +(y1-y2)(y1+y2)/b^2 =0 。 。。(2)
(1)(2) ==> (x1+x2)/a +(y1+y2)/b =0 。。。(3)
点F在直线CD上,直线CD方程: y =(b/a)(x-c), 即:bx-ay =bc
C...全部
A(0,b),B(-a,0),F(c,0),O(0,0)。设:C(x1,y1),D(x2,y2),E(s,t)
Kcd =(y1-y2)/(x1-x2) = Kab =b/a 。。。(1),(K:直线斜率)
C,D两点在椭圆上: x1^2/a^2+y1^2/b^2=1, x2^2/a^2+y2^2/b^2=1
==> (x1-x1)(x1+x2)/a^2 +(y1-y2)(y1+y2)/b^2 =0 。
。。(2)
(1)(2) ==> (x1+x2)/a +(y1+y2)/b =0 。。。(3)
点F在直线CD上,直线CD方程: y =(b/a)(x-c), 即:bx-ay =bc
C,D两点在直线CD上: bx1-ax1 =bc, bx2-ay2 =bc
==> b(x1+x2)-a(y1+y2) =2bc 。
。。(4)
(3)(4) ==> x1+x2 =c, y1+y2 =-bc/a 。。。(5)
OCED是平行四边形:(x1+x2)/2 =(0+s)/2, (y1+y2)/2 =(0+t)/2
==> E(s,t)=E(c,-bc/a)
点E在椭圆上: c^2/a^2+(-bc/a)^2/b^2=1
==> e =c/a = (根号2)/2
点O到直线CD距离 =bc/√(a^2+b^2)
CD长度 =根号[(x1-x2)^2+(y1-y2)^2] =[(x1-x1)/a]*√(a^2+b^2)
√6 = 平行四边形OCED的面积 = 点O到直线CD距离*CD长度
= (x1-x1)bc/a
==> x1-x1 =(a√6)/bc, y1-y2 =(√6)/c 。
。。(6)
(5)^2 +(6)^2 :
2*(x1^2+x2^2) =c^2 +6*(a/bc)^2 。。。(7)
2*(y1^2+y2^2) =(bc/a)^2 +6/c^2 。。。
(8)
(7)/a^2 +(8)/b^2 :
4 = 2*[(x1^2/a^2+y1^2/b^2) +(x2^2/a^2+y2^2/b^2)]
= 2*[(c/a)^2+6/(bc)^2] 。
。。(9)
a^2=b^2+c^2
==> a^2 =(8√3)/3, b^2 =2√3
椭圆方程: x^2/(8√3)/3 +y^2/(2√3) =1
。收起