已知橢圓C的中心在坐標原點,焦點在x軸上,橢圓C上的點到焦點距離的最大值為3,最小值為1.
(1)求橢圓C的標準方程;
(2)D為橢圓C的右頂點,設A是橢圓上異于D的一動點,作AD的垂線交橢圓與點B,求證:直線AB過定點,并求出該定點的坐標.
【答案】
分析:(1)由題設條件可知
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/0.png)
解得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/1.png)
,由此能夠推導出橢圓C的標準方程.
(2)設l:y=kx+m,由方程組
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/2.png)
消去y,得(3+4k
2)x
2+8kmx+4m
2-12=0,然后結合題設條件利用根的判別式和根與系數(shù)的關系求解.
解答:解:(1)由題意設橢圓的標準方程為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/3.png)
,
a+c=3,a-c=1,a=2,c=1,b
2=3,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/4.png)
.
(2)設A(x
1,y
1),B(x
2,y
2),l:y=kx+m,
由
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/5.png)
,得:(3+4k
2)x
2+8mkx+4(m
2-3)=0,
△=64m
2k
2-16(3+4k
2)(m
2-3)>0,3+4k
2-m
2>0
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/6.png)
.
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/7.png)
.
∵AD⊥BD,k
AD•k
BD=-1,(或
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/8.png)
)
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/9.png)
,y
1y
2+x
1x
2-2(x
1+x
2)+4=0,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/10.png)
,7m
2+16mk+4k
2=0,
解得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/11.png)
,且滿足3+4k
2-m
2>0
當m=-2k時,l:y=k(x-2),直線過定點(2,0),與已知矛盾;
當
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/12.png)
時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/13.png)
,直線過定點
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/14.png)
.
綜上可知,直線AB過定點,定點坐標為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181620874247506/SYS201310241816208742475020_DA/15.png)
.
點評:本題綜合考查橢圓的性質(zhì)及應用和直線與橢圓的位置關系,具有較大的難度,解題時要注意的靈活運用.