题干

已知椭圆 C:
x
2
a
2
+
y
2
b
2
=1(a>b>0)
的左,右焦点分别为F1,F2,过F1任作一条与两坐标轴都不垂直的直线,与C交于A,B两点,且△ABF2的周长为8.当直线AB的斜率为
3
4
时,AF2与x轴垂直.

(I)求椭圆C的方程;

(Ⅱ)在x轴上是否存在定点M,总能使MF1平分∠AMB?说明理由.

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答案(点此获取答案解析)

解:(I)由椭圆的定义可知△ABF2的周长4a=8,则a=2,
由直线AB的斜率为
3
4
时,AF2与x轴垂直,则tan∠AF1F2=

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阅读理解

    As we all know,drinking beers while running can upset the stomach.However,that's all part of the attraction of the Beer Mile,a unique race that has participants competing in running and drinking.The Beer Mile Race started off as a tradition in 1990,when a group of Canadians thought it would be fun to race each other while drinking a few beers.But it has come a long way since then.The Beer Mile Race is now a major sporting event with around 100,000 official competitors,brand name sponsors and so on.In 2015,the first Beer Mile World Classic was held in San Francisco,where all of the record holders from Canada and the United States came together for the competition.But this year,the event was organized in London,in an effort to take in more competitors about the sport outside North America.

    The rules of the race are pretty simple: runners have to consume four beers before each of the four 400-meter laps making up the famous Beer Mile.The beers can come in bottles or cans but should not be less than 355ml and must be at least 5% alcohol.The drink like apple juice will not do.Runners must drink the beers within a 10 meter zone before the start/finish line and then they are required to tip the bottle/can over their heads to confirm it is empty.

    In the case of vomiting(呕吐)during the race,and yes,there's quite a bit of that during the Beer.Mile,offenders must complete one more lap at the end of the race.Organizers mention that even in the case of vomiting several times, only one more lap must be completed.

    This year,the Elite Men's race was won by Canadian Corey Bellemore,who also set a new world record with a time of 4 minutes and 34 seconds.“I just drink the beers as quickly as I could,hold them down and get the burps(饱嗝)out,”Corey said.“If you can drink a beer in 8 seconds and also run very fast with a feeling of vomiting,then you'll be fine like me.”