解:(1)因?yàn)?IMG style="VERTICAL-ALIGN: middle" border=0 src="http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523449681216.gif">,
所以
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523449841079.gif)
,即
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/20111124152344984977.gif)
,
當(dāng)m=0時(shí),方程表示兩直線,方程為y=±1;
當(dāng)m=1時(shí),方程表示的是圓;
當(dāng)m>0且m≠1時(shí),方程表示的是橢圓;
當(dāng)m<0時(shí),方程表示的是雙曲線;
(2)當(dāng)
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/20111124152345031941.gif)
時(shí),軌跡E的方程為
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523450461001.gif)
,
設(shè)圓心在原點(diǎn)的圓的一條切線為y=kx+t,
解方程組
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523450621173.gif)
得
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523450781049.gif)
,
即
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523451091183.gif)
,
要使切線與軌跡E恒有兩個(gè)交點(diǎn)A,B,
則使△=
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523451711364.gif)
,
即
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523451871172.gif)
,且
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523452341424.gif)
,
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523453431989.gif)
,
要使
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/20111124152345421970.gif)
,需使
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523454371013.gif)
,
即
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523454841450.gif)
,
所以
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523455311618.gif)
,
所以又因?yàn)橹本€y=kx+t為圓心在原點(diǎn)的圓的一條切線,
所以圓的半徑為
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523455461055.gif)
,
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523455931312.gif)
,
所求的圓為
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523456401004.gif)
;
當(dāng)切線的斜率不存在時(shí),切線為
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523456561002.gif)
,
與
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523456871001.gif)
交于點(diǎn)
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523457501350.gif)
也滿足OA⊥OB;
綜上, 存在圓心在原點(diǎn)的圓
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523457501004.gif)
,使得該圓的任意一條切線與橢圓E恒有兩個(gè)交點(diǎn)A,B,且
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/20111124152345765970.gif)
。
(3)當(dāng)
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/20111124152345796941.gif)
時(shí),軌跡E的方程為
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523458591001.gif)
,
設(shè)直線l的方程為y=kx+t,因?yàn)橹本€l與圓C:
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/20111124152345859979.gif)
(1<R<2)相切于A
1,
由(2)知
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523458901320.gif)
, ①
因?yàn)閘與軌跡E只有一個(gè)公共點(diǎn)B
1,
由(2)知
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523459211173.gif)
得
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523460151049.gif)
,
即
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523460151183.gif)
有唯一解,
則△=
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523460621354.gif)
,
即
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/20111124152346078998.gif)
, ②
由①②得
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523460931330.gif)
,此時(shí)A,B重合為B
1(x
1,y
1)點(diǎn),
由
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523461091424.gif)
中
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/20111124152346140911.gif)
,
所以,
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523462501261.gif)
,
B
1(x
1,y
1)點(diǎn)在橢圓上,所以
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523462651167.gif)
,
所以
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523462651196.gif)
,
在直角三角形OA
1B
1中,
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523463431529.gif)
,
因?yàn)?IMG style="VERTICAL-ALIGN: middle" border=0 src="http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523464211027.gif">當(dāng)且僅當(dāng)
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523464211019.gif)
時(shí)取等號(hào),
所以
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523464371054.gif)
,
即當(dāng)
![](http://thumb.zyjl.cn/pic1/upload/papers/g02/20111124/201111241523464841019.gif)
時(shí),|A
1B
1|取得最大值,最大值為1。