已知椭圆C:x2⼀a2+y2⼀b2=1(a>b>0)的一个顶点为A(2,0),离心率为根号2⼀2,直线y=k(x-1)与椭圆C交于不同的两...

由A(2,0)可得:a=2,
离心率e=c/a=c/2=√2/2,
∴c=√2,
b=√(a^2-c^2)=√2,
∴椭圆方程为:x^2/4+y^2/2=1,
设M(x1,y1),N(x2,y2),
直线方程为:kx-y-k=0,
A点至直线距离h=|2k-0-k|/√(1+k^2)=|k|/√(1+k^2),
x^2/4+k^2(x-1)^2/2=1,
(1+2k^2)x^2-4k^2x+2k^2-4=0,
根据韦达定理,
x1+x2=4k^2/(1+2k^2),
x1*x2=(2k^2-4)/(1+2k^2)
根据弦长公式,
|MN|=√(1+k^2)[(x1+x2)^2-4x1x2]
=√(1+k^2)[16k^4/(1+2k^2)^2-4(2k^2-4)/(1+2k^2)]
=[√(1+k^2)(24k^2+16)]/(1+2k^2)
=[2√(1+k^2)(6k^2+4)]/(1+2k^2)
S△AMN=(1/2)|MN|*h=√[(1+k^2)/(4+6k^2)]*|k|/√(1+k^2)
=√(4+6k^2)|k|/(1+2k^2)=√10/3,
7k^4-2k^2-5=0,
(7k^2+5)(k^2-1)=0,
7k^2+5≠0,
k^2-1=0,
∴k=±1,

由A(2,0)可得:a=2,
离心率e=c/a=c/2=√2/2,
∴c=√2,
b=√(a^2-c^2)=√2,
∴椭圆方程为:x^2/4+y^2/2=1,
设M(x1,y1),N(x2,y2),
直线方程为:kx-y-k=0,
A点至直线距离h=|2k-0-k|/√(1+k^2)=|k|/√(1+k^2),
x^2/4+k^2(x-1)^2/2=1,
(1+2k^2)x^2-4k^2x+2k^2-4=0,
根据韦达定理,
x1+x2=4k^2/(1+2k^2),
x1*x2=(2k^2-4)/(1+2k^2)
根据弦长公式,
|MN|=√(1+k^2)[(x1+x2)^2-4x1x2]
=√(1+k^2)[16k^4/(1+2k^2)^2-4(2k^2-4)/(1+2k^2)]
=[√(1+k^2)(24k^2+16)]/(1+2k^2)
=[2√(1+k^2)(6k^2+4)]/(1+2k^2)
S△AMN=(1/2)|MN|*h=√[(1+k^2)/(4+6k^2)]*|k|/√(1+k^2)
=√(4+6k^2)|k|/(1+2k^2)=√10/3,
7k^4-2k^2-5=0,
(7k^2+5)(k^2-1)=0,
7k^2+5≠0,
k^2-1=0,
∴k=±1,

笨蛋http://ci.baidu.com/YLkSVrvSlN