已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007386319.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150074171081.png)
的焦距為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007417436.png)
,離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007433427.png)
,其右焦點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007449303.png)
,過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007480559.png)
作直線交橢圓于另一點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007495302.png)
.
(Ⅰ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007527654.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007527558.png)
外接圓的方程;
(Ⅱ)若直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007542660.png)
與橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007558373.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007573840.png)
相交于兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007589324.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007620308.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007636820.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007651320.png)
的取值范圍.
試題分析:解: (Ⅰ)由題意知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007901430.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007917669.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007932544.png)
,
解得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007948669.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007979202.png)
橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007386319.png)
的方程為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008010730.png)
2分
由此可得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008026607.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008041606.png)
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008057657.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008073891.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008104736.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008119685.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008135937.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008151628.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150082131140.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008229937.png)
,或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150082441123.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008260625.png)
,或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008275826.png)
4分
①當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007495302.png)
的坐標(biāo)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008307545.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008338849.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007979202.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007527558.png)
外接圓是以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008400300.png)
為圓心,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008416346.png)
為半徑的圓,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007839594.png)
5分
②當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007495302.png)
的坐標(biāo)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008463765.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008478401.png)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008494400.png)
的斜率分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008509212.png)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008525229.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007527558.png)
為直角三角形,其外接圓是以線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008556403.png)
為直徑的圓,圓心坐標(biāo)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008556786.png)
,半徑為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008587823.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008603574.png)
外接圓的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150078541122.png)
綜上可知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007527558.png)
外接圓方程是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007839594.png)
,或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150078541122.png)
7分
(Ⅱ)由題意可知直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008681411.png)
的斜率存在.設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008712642.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008884650.png)
,
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150089151110.png)
得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150089311005.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150089461218.png)
得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008962566.png)
9分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150089771131.png)
…
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015008993856.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150090091033.png)
10分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240150090401684.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015009055563.png)
,結(jié)合(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015009071266.png)
)得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015009087674.png)
12分
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007870715.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015007885702.png)
14分
點(diǎn)評(píng):主要是考查了直線與橢圓的位置關(guān)系的運(yùn)用,屬于中檔題。
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