问题 解答题
已知抛物线C:y2=2px(p>0),其焦点是椭圆mx2+4y2=1的右焦点,且椭圆的离心率为
2
2

(Ⅰ)试求抛物线C的方程;
(Ⅱ)在y轴上截距为2的直线l与抛物线C交于M,N两点,以线段MN为直径的圆过原点,求直线l的方程;
(Ⅲ)若以原点为圆心,以t(t>0)为半径的圆分别交抛物线C上半支和y轴正半轴于A,B两点,直线AB与x轴交于点Q,试用A点的横坐标x0表示点Q的坐标.
答案

(Ⅰ)∵椭圆mx2+4y2=1的离心率为

2
2

1
m
-
1
4
1
m
=
1
2
,∴m=2

∴2x2+4y2=1的右焦点坐标为(

1
2
,0)

∵抛物线C:y2=2px(p>0),其焦点是椭圆mx2+4y2=1的右焦点,

∴抛物线C的方程为y2=2x;

(Ⅱ)由题意,设l的方程为y=kx+2,设M(x1,y1)、N(x2,y2),

直线方程代入抛物线方程可得k2x2+(4k-2)x+4=0,则x1+x2=-

4k-2
k2
,x1x2=
4
k2

∴y1y2=8-

8k-4
k

∵以线段MN为直径的圆过原点,∴

OM
ON
=0

∴x1x2+y1y2=0

4
k2
+8-
8k-4
k
=0

∴k=-1

∴l的方程为y=-x+2,即x+y-2=0;

(Ⅲ)设圆的方程为x2+y2=t,与抛物线方程联立,可得x2+2x-t=0

设A(x0

2x0
),则t=x02+2x0,B(0,x02+2x0

∴直线AB的方程为y-(x02+2x0)=

2
x0
-(x02+2x0)
x0
(x-0)

令y=0,则x=

x03+2x02
x02+2x0-2
x0

∴Q(

x03+2x02
x02+2x0-2
x0
,0)

阅读理解

阅读理解

     ''Mom, can I bake some bread?" We were 15; my best friend, Hanna, and I, determined to try our

hands at creating some beautiful bread.

     "It's not worth the trouble," my mother said. "It takes lots of time and makes a big mess. Our bakery

bread is delicious without all that effort."

     Begging was useless. Mom's "no" meant "No!"

     But several weeks later, opportunity knocked. My parents were going out for the evening. I

immediately invited Hanna to be my partner in bread-baking crime.

     We studied the recipe. That was easy. "Mix oil into flour then beat in four eggs, one at a time, with

remaining sugar and salt."

     We were not good at breaking eggs. I tried to learn from my mother.

     "Gradually add eight cups of flour. When dough (面团) holds together, squeeze it."

     We took turns working like that. "Is the dough 'holding together'?" we asked each other.

     I remembered my neighbor's instructions: "If it's too sticky, add some flour; if too dry, add water."

     We added water. Then more flour. Then more water. By then, the mass of our dough had grown very much.

     "Place dough on floured surface and squeeze till smooth," the recipe instructed.

     We took turns burying our hands in the damp dough, pinching, squeezing, and feeling it leak between

fingers. "Clean and oil bowl, and then return dough to bowl. Cover and let dough rise in a warm place for one hour."

     This was good news-we'd have a break. On the dirty kitchen chairs, we dreamed about our beautiful

bread. "See?" we would tell my mom. "Isn't it worth the work?"

     Hanna and I couldn't help glancing at the rising process every few minutes. But nothing happened.

     "Maybe something will happen in the hot oven," I said.

     Unfortunately, when we removed the loaves from the oven, they were like hard stones.

     Mom was right; it takes time and effort. It sometimes makes a mess. But still it feels good, somehow, to be part of that long, ongoing chain of bread bakers. Since that night, both Hanna and I have learned to do

it right.

1. To the writer, what her mother said was _________.

A. law

B. rubbish

C. advice

D. warning

2. Which of the following can best describe the children's feelings while making their first bread?

A. Confident; hopeful;proud  

B. Curious; hopeful; disappointed

C. Interested; excited; satisfied  

D. Worried; satisfied; proud

3. Which of the following did the writer do without referring to the instructions?

A. Placed dough on floured surface.    

B. Added eight cups of flour to eggs.

C. Returned dough to a cleaned bowl.  

D. Placed the dough into the hot oven.

4. The passage mainly tells us ___________.

A. the process of making bread          

B. the conflict between mother and daughter

C. the first experience of making bread    

D. the way of doing housework

单项选择题