数学椭圆方程问题

椭圆已知椭圆x2/a2+y2/b

解: (1)设A(x1,y1),B(x2,y2),则M((x1+x2)/2,(y1+y2)/2) k1=(y1-y2)/(x1-x2), k2=[(y1+y2)/2]/[(x1+x2)]/2], x1^2/a^2+y1^2/b^2=1 x2^2/a^2+y2^2/b^2=1 两式相减并整理得: (x1+x2)(x1-x2)/a^2=-(y1+y2)(y1-y2)/b^2 即b^2/a^2=-(y1+y2)(y1-y2)/(x1+x2)(x1-x2)=-k1k2 k1k2≥-1/2,则-k1k2≤1/2 即b^2/a^2≤1/2,b^2=a^2-c^2≤(1/2)a^2 得c^2/a^2≥1...全部

  解: (1)设A(x1,y1),B(x2,y2),则M((x1+x2)/2,(y1+y2)/2) k1=(y1-y2)/(x1-x2), k2=[(y1+y2)/2]/[(x1+x2)]/2], x1^2/a^2+y1^2/b^2=1 x2^2/a^2+y2^2/b^2=1 两式相减并整理得: (x1+x2)(x1-x2)/a^2=-(y1+y2)(y1-y2)/b^2 即b^2/a^2=-(y1+y2)(y1-y2)/(x1+x2)(x1-x2)=-k1k2 k1k2≥-1/2,则-k1k2≤1/2 即b^2/a^2≤1/2,b^2=a^2-c^2≤(1/2)a^2 得c^2/a^2≥1/2 √2/2≤e<1 即圆心率e的取值范围(√2/2,1) (2)设直线AB的方程:y=k(x-c),代入椭圆方程并整理得 (a^2k^2+b^2)x^2-2a^2k^2cx+a^2k^2c^2-a^2b^2=0 x1+x2=2a^2k^2c/(a^2k^2+b^2),x1x2=(a^2k^2c^2-a^2b^2)/(a^2k^2+b^2) (x1-x2)^2=(x1+x2)^2-4x1x2=4a^2b^4(k^2+1)/(a^2k^2+b^2)^2 弦AB的长|AB|=√[(x1-x2)^2+(y1-y2)^2]=√(k^2+1)(x1-x2)^2 =√4a^2b^4(k^2+1)^2=2ab^2(k^2+1)/(a^2k^2+b^2) 原点到直线AB的距离d=|kc|/√(k^2+1) 平行四边形OACB的面积=|AB|*d=[2ab^2(k^2+1)/(a^2k^2+b^2)]|kc|/√(k^2+1)=3ac√5/10 解得: 。
  收起