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问题 解答题
已知椭圆 
x2
4
+y2=1的左顶点为A，过A作两条互相垂直的弦AM、AN交椭圆于M、N两点．
（1）当直线AM的斜率为1时，求点M的坐标；
（2）当直线AM的斜率变化时，直线MN是否过x轴上的一定点，若过定点，请给出证明，并求出该定点，若不过定点，请说明理由．
答案

（1）直线AM的斜率为1时，直线AM：y=x+2，（1分）

代入椭圆方程并化简得：5x2+16x+12=0，（2分）

解之得x1=-2，x2=-

6
5
，∴M(-
6
5
，
4
5
)．（4分）

（2）设直线AM的斜率为k，则AM：y=k（x+2），

则

y=k(x+2)
x2
4
+y2=1
化简得：（1+4k2）x2+16k2x+16k2-4=0．（6分）

∵此方程有一根为-2，∴xM=

2-8k2
1+4k2
，（7分）

同理可得xN=

2k2-8
k2+4
．（8分）

由（1）知若存在定点，则此点必为P(-

6
5
，0)．（9分）

∵kMP=

yM
xM+
6
5
=
k(
2-8k2
1+4k2
+2)
2-8k2
1+4k2
+
6
5
=
5k
4-4k2
，（11分）

同理可计算得kPN=

5k
4-4k2
．（13分）

∴直线MN过x轴上的一定点P(-

6
5
，0)．（16分）

解答题

妞妞在商店里看到一块漂亮的方桌布，非常想买，但当她拿起来时感觉桌布不太方，商店老板看她犹豫的样子，马上过来折叠起一组对角，让妞妞看这一组对角是否重合，妞妞还有些疑惑，老板又折叠起另一组对角，让妞妞检验，妞妞终于买了这块桌布，请你判断：

（1）妞妞买的这块桌布，是正方的吗？

（2）你认为应该如何检验，才能又快又准确呢？

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单项选择题

Millions of Americans and foreigners see GI.Joe as a mindless war toy, the symbol of American military adventurism, but that’ s not how it used to be. To the men and women who (1) in World War I1 and the people they liberated, the GI. was the (2) man grown into hero, the pool farm kid torn away from his home, the guy who (3) all the burdens of battle, who slept in cold foxholes, who went without the (4) of 1hod and shelter, who stuck it out and drove back the Nazi reign of murder. This was not a vohmteer soldier, not someone well paid, (5) an average guy, up (6) the best trained, best equipped, fiercest, most brutal enemies seen in centuries.

His name is not much. GI. is just a military abbreviation (7) Govermnent Issue, and it was on all of the article (8) to soldiers. And Joe A common name for a guy who never (9) it to the top. Joe Blow, Joe Magrae... a working class name.The United States has (10) had a president or vieepresident or secretary of state Joe.

GI. Joe had a (11) career fighting Geman, Japanese, and Korean troops. He appears as a character, or a (12) of American personalities, in the 1945 movie The Story of GL Joe, based on the last days of war correspondent Ernie Pyle. Some of the soldiers Pyle (13) portrayde themselves in the fihn. Pyle was famous for covering the (14) side of the warl, writing about the dirt-snow-and-mud soldiers, not how many miles were (15) or what towns were captured or liberated. His reports (16) the "willie" cartoons of famed Stars and Stripes artist Bill Maulden. Both men (17) the dirt and exhaustion of war, the (18) of civilization that the soldiers shared with each other and the civilians: coffee, tobacco, whiskey, shelter, sleep. (19) Egypt, France, and a dozen more countries, G. I. Joe was any American soldier, (20) the most important person in their lives.

8（）

A.handed out

B.turn over

C.brought back

D.passed down
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