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问题 解答题
已知双曲线C：
x2
a2
-
y2
b2
=1（a＞0，b＞0）的右准线与一条渐近线交于点M，F是右焦点，若|MF|=1，且双曲线C的离心率e=
6
2
．
（1）求双曲线C的方程；
（2）过点A（0，1）的直线l与双曲线C的右支交于不同两点P、Q，且P在A、Q之间，若
AP
=λ
AQ
且λ≥
1
3
，求直线l斜率k的取值范围．
答案

（1）由对称性，不妨设M是右准线x=

a2
c
与一渐近线y=
b
a
x的交点，

其坐标为M（

a2
c
，
ab
c
），∵|MF|=1，∴
b4
c2
+
a2b2
c2
=1，

又e=

c
a
=
6
2
∴
b
a
=
e2-1
=
2
2
，c2=a2+b2=
3
2
a2，

解得a2=2，b2=1，所以双曲线C的方程是

x2
2
-y2=1；（6分）

（2）设直线l的斜率为k，则l的方程为y=kx+1，设点P（x1，y1），Q（x2，y2），

由

y=kx+1
x2-2y2=2
得：（1-2k2）x2-4kx-4=0，

∵l与双曲线C的右支交于不同的两点P、Q，

∴

△=16k2+16(1-2k2)＞0
x1+x2=
-4k
2k2-1
＞0
x1x2=
4
2k2-1
＞0
1-2k2≠0

∴

1
2
＜k2＜1且k＜0①（9分）

又∵

AP
=λ
AQ
且P在A、Q之间，λ≥
1
3
，∴x1=λx2且
1
3
≤λ＜1，

∴

(1+λ)x2=
-4k
2k2-1
λ
x22
=
4
2k2-1
∴
(1+λ)2
λ
=
4k2
2k2-1
=2+
2
2k2-1
，

∵f(λ)=

(1+λ)2
λ
=λ+
1
λ
+2在[
1
3
，1)上是减函数（∵f′（λ）＜0），

∴4＜f(λ)≤

16
3
，

∴4＜2+

2
2k2-1
≤
16
3
，由于k2＞
1
2
，∴
4
5
≤k2＜1②（12分）

由①②可得：-1＜k≤-

2
5
5
，（13分）

即直线l斜率取值范围为(-1，-

2
5
5
]（14分）

问答题

The importance of agriculture cannot be overstated. Mom than 50 percent of the world’s labor force is employed in agriculture. The distribution in the early 1980s ranged from 67 percent of those employed in Africa to less than 5 percent in North America. In Western Europe, the figure was about 16 percent; in Eastern Europe and the Soviet Union, about 32 percent; and in Asia, about 68 percent. Farm size varies widely from region to region. Recently the average for Canadian farms was about 186 ha (about 460 acres) per farm, and for U.S. farms, about 175 ha (about 432 acres). The average size of a single landholding in the Philippines, however, may be somewhat less than 3.6 ha (less than 9 acres), and in Indonesia, a little less than 1.2 ha (less than 3 acres). Size also depends on the purpose of the farm. Commercial farming, or production for cash, is usually done on large holdings. The plantations of Latin America are large, privately owned estates worked by tenant labor. Single-crop plantations produce tea, rubber, cocoa. Wheat farms are most efficient when they comprise some thousands of hectares and can be worked by teams of people and machines. Australian sheep stations and other livestock farms must be large to provide grazing for thousands of animals. Individual subsistence farms or small-family mixed-farm operations are decreasing in number in developed countries but are still numerous in the developing countries of Africa and Asia. A "back- to-the-land" movement in the U.S. reversed the decline of small farms in New England and Alaska in the decade from 1970 to 1980. The conditions that determine what will be raised in an area include climate, water supply, and terrain. Over the 10,000 years since agriculture began to be developed, peoples everywhere have discovered the food value of wild plants and animals and domesticated and bred them. The most important are cereals such as wheat, rice, barley, com and rye. Agricultural income is also derived from non-food crops such as rubber, fiber plants, tobacco, and oilseeds used in synthetic chemical compounds. Money is also derived from raising animals for pelt. Much of the foreign exchange earned by a country may be derived from a single commodity; for example, Sri Lanka depends on tea, Denmark specializes in dairy products, Australia in wool, and New Zealand and Argentina in meat products. In the U.S., wheat has become a major foreign exchange commodity in recent years. The importance of an individual country as an exporter of agricultural products depends on many variables. Among them is the possibility that the country is too little developed industrially to produce manufactured goods in sufficient quantity or technical sophistication. Such agricultural exporters include Ghana with cocoa, and Myanmar with rice. On the other hand, an exceptionally well-developed country may produce surpluses not needed by its own population; this has been true of the U.S., Canada, and some of the West European countries.
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