问题 解答题

在直角坐标系xoy中,圆O的方程为x2+y2=1.

(1)若直线l与圆O切于第一象限且与坐标轴交于点A,B,当|AB|最小时,求直线l的方程;

(2)若A,B是圆O与x轴的交点,C是圆在直径AB的上方的任意一点,过该点作CD⊥AB交圆O于点D,当点C在圆O上移动时,求证:∠OCD的角平分线经过圆O上的一个定点,并求出该定点的坐标.

答案

(1)设直线l的方程为

x
a
+
y
b
=1(a>0,b>0),则
1
1
a2
+
1
b2
=1

1
a2
+
1
b2
=1,∴ab≥2(当且仅当a=b=
2
时,取等号)

∴|AB|=

a2+b2
2ab
≥2(当且仅当a=b=
2
时,取等号)

即|AB|最小为2,此时直线l的方程为x+y-

2
=0;

(2)证明:设∠OCD的角平分线为CP,交圆于P,则∠OCP=∠DCP

因为OC、OP为圆的半径,所以∠OCP=∠OPC,所以∠DCP=∠OPC

所以CDOP

因为CD⊥AB,A、B为定点,所以OP⊥AB

所以P为定点,坐标为(0,-1)

阅读理解
阅读理解。
                                                          An Experiment
Materials: Newspaper, "ACE" hardware ruler (1/8" thick), a flat table.
Purposes: We'll show that there is air pressure pushing on us, from every direction while we're on this Earth.
Procedure:
     First put a thin ruler on a flat table with a little
less than half of it hanging off the edge of the table
(see the drawing below). Next place a sheet of
newspaper over the ruler flat against the table.
Make sure to have as little air as possible under the
   
paper so that the fold line of the newspaper is at the ruler. Then quickly strike the end of the ruler hanging off
the edge of the table. If you strike it quick enough, the ruler will break near the table edge.
What's going on?
     The Earth is covered in a layer of air that is nearly 80 miles thick and at sea level (the bottom) exerts or
"pushes" hard almost 15 pounds of pressure per square inch. That means that a full sheet of newspaper laid out
flat has nearly 9,300 pounds of air above it.
     When you break the ruler above, you are able to break it because of the "heavy" air pushing down on the
paper while you quickly strike the ruler. At first, the table is pushing back on the paper, and if you move the
ruler quick enough, other air around the edges of the paper can't get under the paper fast enough, so you are
trying to lift that 9,300 pounds with the ruler! Some air gets under the paper, but not enough, so the ruler
breaks.
1. By doing the experiment, we may realize _____.
[ ]
A. that air exits everywhere 
B. why there are directions
C. that we live with air pressure
D. how air helps on the earth
2. How many steps are there in this experiment?
[ ]
A. 2
B. 3
C. 4
D. 5
3. The underlined word "exert" probably means _____.
[ ]
A. cover completely
B. advocate actively
C. influence gradually
D. press heavily
4. The ruler breaks under a quick strike mainly because of _____.
[ ]
A. the air pressure on the newspaper
B. the heavy weight of the newspaper
C. the heavy weight of the flat table
D. the strong power used on the ruler
单项选择题