问题 解答题
已知双曲线C:
x2
a2
-
y2
b2
=1(a>0,b>0)的左、右焦点分别为F1,F2,离心率为3,直线y=2与C的两个交点间的距离为
6

(I)求a,b;
(II)设过F2的直线l与C的左、右两支分别相交于A、B两点,且|AF1|=|BF1|,证明:|AF2|、|AB|、|BF2|成等比数列.
答案

(I)由题设知

c
a
=3,即
b2+a2
a2
=9,故b2=8a2

所以C的方程为8x2-y2=8a2

将y=2代入上式,并求得x=±

a2+
1
2

由题设知,2

a2+
1
2
=
6
,解得a2=1

所以a=1,b=2

2

(II)由(I)知,F1(-3,0),F2(3,0),C的方程为8x2-y2=8    ①

由题意,可设l的方程为y=k(x-3),|k|<2

2
代入①并化简得(k2-8)x2-6k2x+9k2+8=0

设A(x1,y1),B(x2,y2),

则x1≤-1,x2≥1,x1+x2=

6k2
k2-8
x1x2=
9k2+8
k2-8
,于是

|AF1|=

(x1+3)2+y1 2
=
(x1+3)2+8x1 2-8
=-(3x1+1),

|BF1|=

(x2+3)2+y2 2
=
(x2+3)2+8x2 2-8
=3x2+1,

|AF1|=|BF1|得-(3x1+1)=3x2+1,即x1+x2=-

2
3

6k2
k2-8
=-
2
3
,解得k2=
4
5
,从而x1x2=
9k2+8
k2-8
=-
19
9

由于|AF2|=

(x1-3)2+y1 2
=
(x1+3)2+8x1 2-8
=1-3x1

|BF2|=

(x2-3)2+y2 2
=
(x2+3)2+8x2 2-8
=3x2-1,

故|AB|=|AF2|-|BF2|=2-3(x1+x2)=4,|AF2||BF2|=3(x1+x2)-9x1x2-1=16

因而|AF2||BF2|=|AB|2,所以|AF2|、|AB|、|BF2|成等比数列

判断题
单项选择题

Oxford and Cambridge University Boat Clubs have both taken the opportunity to travel to Spain this month to train in less testing weather conditions than those which have, quite literally, already blown around the UK in January of 2007.

Each group of athletes has been focused on training to the maximum, working on technique and molding themselves into two potential fighting units per Club for the 2007 Boat Race, sponsored by Xchanging. One set will be in the Blue Boat for each club and one set will race as reserves in Isis, for Oxford, and Goldie, for Cambridge.

In these modern times, the Head Coach for each club has a huge input on selection even though the crew is still named by the President. Just twenty years ago the balance was not quite the same.

The year 1987 will always be remembered in the history of this great Race as the year of the "mutiny" at Oxford. It is a tale which has since been retold and reworked in both a book and a movie.

This was the season for which mature Scottish student, Donald Macdonald, was elected President, having all ready won a Blue in 1986. Macdonald re-appointed Daniel Topolski (now a renowned rowing journalist and broadcaster) as Chief Coach.

Part of the 1986/87 squad at Oxford included American Chris Clark, now a coach at an American University, and four fellow US internationals.

Allegedly, a split appeared in the squad between the American quintet, all experienced and leading oarsmen, and those rowers closest to Macdonald. The Scot’s group were happy to follow Topolski’s regime whilst the others were not so sure.

Following a contentious seat racing trial in January of 1987, Topolski decided to move Clark to the bow-side of the beat. Clark disagreed. Topolski held firm. As a result Clark and his "group" within the squad decided not to ROW and sought a takeover.

The squabble was played out extensively in the UK national media and caught the public imagination. MacDonald sought support from the college captains and eventually won a vote of confidence by 28 votes to 11.

Without the Americans, the Oxford crew was immediately considered a lost cause. Cambridge were overwhelming favorites to win.

As it turned out, though, this was a Race which would prove why sport, and particularly The Boat Race, can be so fascinating.

According to the passage, in crew-naming, the President ().

A. has no influence in the Head Coach’s opinion

B. gives little care to the idea of the Head Coach

C. used to be the only one to make decisions

D. had greater say 20 years ago than today