问题 解答题
已知椭圆
x2
a2
+
y2
b2
=1(a>b>0)
的离心率e=
6
3
,过点A(0,-b)和B(a,0)的直线与原点的距离为
3
2

(1)求椭圆的方程;
(2)已知定点E(-1,0),若直线y=kx+2(k≠0)与椭圆交于C、D两点,问:是否存在k的值,使以CD为直径的圆过E点?请说明理由.
答案

(1)直线AB方程为bx-ay-ab=0,

依题意可得:

c
a
=
6
3
ab
a2+b2
=
3
2

解得:a2=3,b=1,

∴椭圆的方程为

x2
3
+y2=1.

(2)假设存在这样的值.

y=kx+2
x2+3y2-3=0

得(1+3k2)x2+12kx+9=0,

∴△=(12k)2-36(1+3k2)>0…①,

设C(x1,y1),D(x2,y2),

x1+x2=-
12k
1+3k2
x1x2=
9
1+3k2
…②

而y1•y2=(kx1+2)(kx2+2)=k2x1x2+2k(x1+x2)+4,

要使以CD为直径的圆过点E(-1,0),

当且仅当CE⊥DE时,

则y1x1+y2x2+1=-1,

即y1y2+(x1+1)(x2+1)=0,

∴(k2+1)x1x2+(2k+1)(x1+x1)+5=0…③

将②代入③整理得k=

7
6

经验证k=

7
6
使得①成立综上可知,存在k=
7
6
使得以CD为直径的圆过点E.

阅读理解

Marlin Perkins’ keen interest in animals, especially snakes, brought him a lot of trouble when he was young. His landlady never excused him for keeping snakes in his closet. After two years of college, he left school and found a job in a zoo, sweeping the sidewalks for $3.50 a day. He was very happy about it because he could now watch animals and study them all day long. He soon became an official in charge of reptiles (爬行动物)and then director of the Buffalo Zoo.

Through his snake-hunting expeditions, Perkins won his fame as a snake expert. His love for snakes is such that he thinks of them in terms of (根据) personalities. However, snakes do not have the universal appeal as he wishes. Even Perkins’ most favorite animal Bushman, the 550-pound toughest African ape, was frightened by snakes. Bushman once escaped from his cage. Nothing could draw him back. Finally a keeper went to the reptile house, got three little snakes, and dangled (摇晃) them toward Bushman. The great ape nearly tore the place apart getting back to the safety of his cage.

Perkins likes all animals and takes more interest in them than many parents take in their own children. He keeps daily records on all the baby animals, takes their temperature and watches their diet strictly. Speaking about his success, Perkins sums up his philosophy of life as follows: “My life has taught me that the surest path to success is to find out what you most enjoy doing, then set out to do it.”

40.Perkins’ first zoo job was __________.

A. collecting snakes     B. sweeping sidewalks

C. looking after an ape   D. taking notes on animals

41.Bushman’s reaction (反映) suggests that ___________.

A. he was easy to be nervous   B. most people dislike animals

even animals dislike snakes  D. he was a tough animal to manage

42.Perkins believes the best way to success is to __________.

A. do what you like most     B.work hard at it

C.do what people don’t like   D.be brave enough

43. According to the passage, Marlin Perkins entered his life work through ______.

A. help      B.chance        C.choice      D.necessity

44.A good title for the passage would be ________.

A.Lovely Snakes                B.A Good Expert

C.Marlin Perkins’ world          D.Human’s Friend---Animal

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