问题 填空题
已知点M是椭圆
x2
a2
+
y2
b2
=1(a>b>0)
上的一点,过M作斜率分别为k1,k2的直线,交椭圆于A,B两点,且A,B关于原点对称,则k1k2=-
b2
a2
.类比椭圆的这个性质,设M是双曲线
x2
a2
-
y2
b2
=1(a>0,b>0)
上的一点,过M作斜率分别为k1,k2的直线,交双曲线于A,B两点,且A,B关于原点对称,则k1•k2=______.
答案

设点M半短轴上的顶点,则M作斜率分别为k1,k2的直线,交椭圆于A,B两点,且A,B关于原点对称,

设A和B两点坐标为(a,0),(-a,0),即k1=

b
a
,k2=-
b
a
k1k2=-
b2
a2

类比椭圆性质类推双曲线的性质,

设点M实轴上顶点(a,0),则M作斜率分别为k1,k2的直线,交椭圆于A,B两点,且A,B关于原点对称,

设A和B两点坐标为为(x,y),(-x,-y),

即k1=x+a,k2=

y
x-a
,k1•k2=
y
x+a
y
x-a
=
y2
x2-a2

结合

x2
a2
-
y2
b2
=1化简可得k1•k2=
b2
a2

故答案为

b2
a2

单项选择题

Passage One

The increase in leisure time, the higher standard of living, the availability of cars to a wider range of the population and, perhaps, a broadening of personal horizons have all contributed to a drastic change in the summer week-end habits of the British publiC. Now, on most Saturdays in the months loosely called summer, it is possible to see family saloons loaded with picnics and crammed to bursting with several generations of pleasure-bent Smiths’. Like competitors in some grossly disorganized rally, they nose their way through the neat drab streets of council estates, converging on the main roads, then crawl as best they can out into the open country and towards the coast.

Congestion and the frustration of wasting precious time at the receiving end of someone else’s exhaust fumes gets the pursuit of enjoyment off to had start; tempersbecome frayed. Children, traditionally the target for fathers’ ill-humor, are singled out for special treatment. The past week’s misdeeds are unearthed and magnified out of all reasonable proportion; mothers leap to their broods’ defense and, before long, vows that never again will this outing be repeated are being hurled back and forth. Of course, by this time, the children have wisely extracted themselves from the argument and are quietly amusing themselves by looking at their irate elders or gaping at the unfamiliar sight of animals in fields, often so much stranger to them than the corresponding naked shapes they are wont to see in butchers’ windows.

Eventually, tempers partially restored, the sea is in sight. The paraphernalia of enjoyment is set up on teeming beach, sand mysteriously appears in every sandwich, pale industrial legs are exposed in self-conscious nakedness.

The children drift away, quite capable of finding enough magic in this exciting, watery world to occupy them fully until they are gathered in again. Fathers and mothers, and quite possibly some members of a previous generation, settle back to receive the sun and dream away the tensions brought to a climax by the journey. Fathers eye with furtive lustfulness and mothers glare with disapproval and envy as the shapely matrons of tomorrow splash and play and race coquettishly around them, spraying water and sand and disturbing any hopes of peace.

At length the shadows drop and chill in the air brings an end to the idyll. The lobster skin is painfully covered up and the day’s debris half-heartedly collected. The family is rounded up and the brief dreams trodden into the sand along with the wasted paper.

When they prepare to leave, they ().

A.carefully pack away the lobsters they have caught

B.put plasters on the places where they have been hurt

C.cover over their debris with waste paper

D.dress carefully to avoid any further irritation

选择题