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问题 解答题
过双曲线
y2
3
-x2=1的上支上一点P作双曲线的切线分别交两条渐近线于点A，B．（1）求证：
OA
•
OB
为定值．（2）若
OB
=
AM
，求动点M的轨迹方程．
答案

解．（1）∵双曲线

y2
3
-x2=1的上支可表示为函数y=
3+3x2
，且y′=
1
2
×
6x
3+3x2
=
3x
3+3x2

设P（x0，y0）是双曲线上任一点，则双曲线在该点处的切线为y-y0=

3x 0
3+3x 02
（x-x0）

即y-y0=

3x 0
y0
（x-x0），即y0y-3x0x=3，

与渐近线方程y=

3
x联立，解得A(
3
y0-
3
x0
，
3
y0-
3
x0
)（由于P不在双曲线的渐近线上，故y0±
3
x0≠0）；

与渐近线y=-

3
x联立，解得B(
-
3
y0+
3
x0
，
3
y0+
3
x0
)，

∴

OA
•
OB
=
-3
y20
-3
x20
+
9
y20
-3
x20
=
-3
3
+
9
3
=2（定值）

（2）设M（x，y）为所求轨迹上一点，由

OB
=
AM
知
OM
=
OA
+
OB
，由（1）有

x=
3
y0-
3
x0
+
-
3
y0+
3
x0
y=
3
y0-
3
x0
+
3
y0+
3
x0

即

x0=
x
2
y0=
y
2

再由P（x0，y0）在双曲线

y2
3
-x2=1 （y＞0）上

∴

y20
3
-
x20
=1，

∴

y2
4
3
-
x2
4
=1

故所求轨迹为

y2
12
-
x2
4
=1(y＞0)．

单项选择题

某人为改善居住条件将已居住4年的住房出售，该房评估价格为37万元，缴纳的相关税费为5万元，售价为62万元。应缴纳的土地增值税是( )万元。

A．3.2
B．4.8
C．6
D．8.4
查看答案
单项选择题

It sounds like a science fiction, but researchers say it’s a scientific fact: Microscopic organisms dubbed "killer algae" are paralyzing fish with unknown toxins and then eating away at their flesh. They might be making people sick, too.
The name given to this single-cell organism, or dinoflagellate, is Pfiesteria piscimorte- literally, "fish killer. " It was discovered at North Carolina State University in 1988, but at the same time, few scientists believed in its existence, much less in its highly unusual predatory nature. But ongoing research has led to international acknowledgment of the phenomenon and, recently, research funding.
Seemingly prompted by an unknown substance secreted by fish, the aggressive creature swims into action. It sends neurotoxins into the water and air, paralyzing a fish’s nervous system, and causing it to gasp for air at the surface. Eventually the fish suffocates. The killer dinoflagellate then attaches itself to the fish and begins sucking away at its flesh. This macabre scenario may help solve several sea mysteries, such as why fishermen report seeing "fish walks" (fish trying to leave the water) and pools of dead fish with holes eaten through them.
Whether humans are affected by the dinoflagellate’s toxin remains to be seen. Researchers think it’s unlikely eating fresh fish are in danger, but anyone with frequent exposure to the creature could be in peril. Several researchers working with the algae have reported bouts of memory loss and disorientation.
To understand the killer algae further, scientists must fully characterize their toxin and determine what stimulates them to attack. They also need to know if the algae are hurting fish populations, and whether pollutants make them more likely to attack.

The passage is primarily concerned with ______.

A. informing readers of different sea creatures
B. describing predatory microscopic organisms
C. solving some long-lasting mysteries about the sea
D. discussing the environment for various fishes
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