问题 解答题

设实数x,y同时满足条件:4x2-9y2=36,且xy<0.

(1)求函数y=f(x)的解析式和定义域;

(2)判断函数y=f(x)的奇偶性;

(3)若方程f(x)=k(x-1)(k∈R)恰有两个不同的实数根,求k的取值范围.

答案

(1)∵4x2-9y2=36,

y=±

2
3
x2-9

∵xy<0,∴y≠0.

又∵4x2-36=9y2>0,

∴x>3,x<-3.

∵xy<0,

f(x)=

2
3
x2-9
(x<-3)
-
2
3
x2-9
(x>3)

函数y=f(x)的定义域为集合D={x∈R|x>3,x<-3}.

(2)当x<-3有-x>3,f(-x)=-

2
3
(-x)2-9
=-
2
3
x2-9
=-f(x),

同理,当x>3时,有f(-x)=-f(x).

任设x∈D,有f(-x)=-f(x),

∴f(x)为定义域上的奇函数.

(3)联立方程组

4x2-9y2=36
y=k(x-1)

可得,(4-9k2)x2+18k2x-(9k2+36)=0,

(Ⅰ)当k2=

4
9
时,即k=±
2
3
时,方程只有唯一解,与题意不符;

k≠±

2
3

(Ⅱ)当k2

4
9
时,即方程为一个一元二次方程,

要使方程有两个相异实数根,

则△=(18k22+4×(4-9k2)(9k2+36)>0.

解之得  -

2
2
<k<
2
2
,但由于函数f(x)的图象在第二、四象限.

故直线的斜率k<0,

综上可知-

2
2
<k<-
2
3
-
2
3
<k<0

多项选择题
阅读理解
阅读理解
     Have you ever been to London? Here is some hotel information for you.
     Kingsway Hall Hotel
     We offer quality service (优质服务) in Kingsway Hall Hotel. This location puts guests in easy reach of many places, such as the British Museum and Piccadilly Circus. Our hotel also has modern Western food, and offers a perfect dining environment. The lounge bar is a comfortable place to enjoy a drink while
watching the tourists. There are hundreds of dining choices within walking distance. Non-smoking rooms
are ready for guests.
     Hilton Docklands Riverside
     Located on the bank of the Thames, our hotel is about two miles east of Tower Bridge and 36 miles
from London Heathrow Airport. Our hotel has a beauty salon, a flower shop and a swimming pool. The
hotel also offers laundry (洗衣店) services, babysitting, shining shoes, and local area transportation.
     Windermere Hotel
     Located in the center of London, our hotel puts guests within easy reach of London's well-known
buildings. Buckingham Palace and the Royal Parks are just a short walk away. Within 1.5 miles you can
visit Westminster Abbey and the Houses of Parliament. The theatres, Harrods shopping centre, and the
designer shops of Sloane Square and Knightsbridge are all only a short walk away or a taxi ride away.
1. What does Kingsway Hall Hotel provide for its guests?
A. Chinese food.
B. A lounge bar.
C. Smoking rooms.
D. A swimming pool.
2. What is special about Hilton Docklands Riverside?
A. It offers different kinds of service.
B. It offers a reasonable price.
C. It has convenient transport.    
D. It has delicious food.
3. This passage is most likely to be found in ______.
A. a travel guide
B. a story book
C. a technical report
D. a geography magazine