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问题 解答题
已知椭圆的中心在原点，焦点在x轴上，离心率为
3
2
，过点M（-1，0）的直线l与椭圆交于P、Q两点．
（1）若直线l的斜率为1，且
PM
=-
3
5
QM
，求椭圆的标准方程；
（2）若（1）中椭圆的右顶点为A，直线l的倾斜角为α，问α为何值时，
AP
•
AQ
取得最大值，并求出这个最大值．
答案

（1）e=

3
2
⇒
c2
a2
=
3
4
⇒a2=4b2，故椭圆方程为x2+4y2=4b2，

设P（x1，y1）、Q（x2，y2），由

PM
=-
3
5
QM
得y1=-
3
5
y2，

由

y=x+1
x2+4y2=4b2
消去x得5y2-2y+1-4b2=0，∴y1+y2=
2
5
，y1y2=
1-4b2
5
，

由此得b2=1，a2=4，椭圆方程为

x2
4
+y2=1；

（2）当直线l的斜率存在时，设l的方程为：y=k（x+1）代入椭圆方程得：x2+4k2（x+1）2=4⇒（1+4k2）x2+8k2x+4k2-4=0⇒

x1+x2=-
8k2
1+4k2
x1x2=
4k2-4
1+4k2
，所以
AP
•
AQ
=(x1-2，y1)•(x2-2，y2)=(x1-2)(x2-2)+y1y2=（1+k2）x1x2+（k2-2）（x1+x2）+4+k2=
33k2
1+4k2
=
33
1
k2
+4
＜
33
4
，

当直线l的斜率不存在即α=90°时，

AP
•
AQ
=
33
4
，

因此当α=90°时，

AP
•
AQ
取得最大值，最大值为
33
4

计算题

（14分）在大型文艺节目《白云之帆》表演中，由地面上近1200人用绳索拉着180名演员上演着“空中飞人”。九分钟的时间里，180名塔沟武校的武林小子在帆屏上时而俯冲而下，时而攀沿升空，演绎了一场世界罕见的惊险、刺激却又浪漫、温馨的节目。现在把他们某次训练过程中的情节简化成如下模型：如图所示，地面上的人通过定滑轮用轻绳将质量m=50kg的演员从静止竖直向上拉高28m，用时t=7s，演员在竖直方向上先做匀加速直线运动，再做匀减速直线运动后静止在高空。其加速和减速过程中的加速度之比为2:5，忽略滑轮的质量和滑轮与轴之间的摩擦力。求：

（1）演员在匀加速上升过程中的加速度；

（2）演员在匀减速上升过程中绳子对地面上的人的拉力为多少？（g=10m/s2）
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Less education, income linked to obesity in women, not men


Women who are better educated and live in households that are middle-income or above are less likely to be obese than women who are less educated and live in the lowest income households, new government research shows.
Among men, there is not a statistically significant (26) in obesity based on income and very (27) difference based on education, the data show.
(28) , about one in three U.S. adults-almost 73 million people-are obese, which is (29) 30 or more pounds over a (30) weight. Extra weight raises the (31) of diabetes, heart disease, some types of cancer and other (32) .
"There is a relationship between obesity and income, but it’s not a (33) story," says Cynthia Ogden, an epidemiologist with the National Center for Health Statistics, part of the Centers for Disease Control and Prevention.
"When looking at these two (34) of socioeconomic status-income and education-their (35) is greater on women than men," she says.
Jennifer Lovejoy, president of the Obesity Society, says that (36) -income women may be more likely to become obese because of environmental (37) such as lack of access to safe places to do physical activity and easy access to fast food.
Among the findings:
·29% of women who live in households with an annual income of $77,000 or more for a family of four are obese in opposition (38) 42% of women who live in households with an annual income below $29,000 for a family of four.
·23% of women with a college degree are obese, significantly less than the 42% of women with (39) than a high school education.
·33% of men who live in households with an annual income of $77,000 or more for a family of four are obese, (40) 29% of men who live in households with an annual income below $29,000 for a family of four are obese. This difference is not considered statistically (41) .
The analysis is based on (42) from the National Health and Nutrition Examination Survey, which is considered the gold (43) for evaluating obesity because it is a(an) (44) survey of people whose weight and height are actually (45) rather than being self-reported.

A. reliableB. poor C. informative D. significant
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