问题 解答题

抛物线y2=2px的焦点弦AB的中点为M,A、B、M在准线上的

影依次为C、D、N.求证:

(1)A、O、D三点共线,B、O、C三点共线;

(2)FN⊥AB(F为抛物线的焦点)

答案

(1)设A(x1,y1)、B(x2,y2)、中点M(x0,y0),焦点F的坐标是(,0).

得ky2-2py-kp2=0.

∴A、B、M在准线上的射影依次为C、D、N,

∴C(-,y1)、D(-,y2)、N(-,y0).

由ky2-2py-kp2=0

得y1y2=-p2

∴kOA=kOD,∴A、O、D三点共线.同理可证B、O、C三点共线.

(2)kFN,当x1=x2时,显然FN⊥AB;当x1≠x2时,

kAB

,∴kFN·kAB=-1.∴FN⊥AB.综上所述知FN⊥AB成立.

同答案

单项选择题
阅读理解

One day, Tomas Jefferson, the president of the USA, was crossing a river with a group of travelers. The river was overflowing its banks. So it was very dangerous to cross it on a horse. Each man was fighting for his life! Then another traveler came and he wanted to cross it, too. But he didn’t have a horse. He watched the group and then asked Jefferson to take him. Jefferson agreed immediately. The man climbed on. After they were both safe to the other side of the river, someone asked the man, “Why did you ask present for help? Did you know him?” The man was shocked because he didn’t know Jefferson was the president. “All I know,” he said, “on some of your faces were written the answer NO, but there was the answer YES on his face.”

小题1:Who was Tomas Jefferson?

A. He was a businessman.                B. He was a teacher.

C. He was a captain.                    D. He was the president of the USA.

小题2: How did they cross the river?

A.In a boat.

B.On their horses.

C.By swimming.

D.By building a bridge.小题3: When the group were crossing the river, ________.

A.water was flowing out of the river

B.it was windy and very cold

C.there were many empty boats in the river

D.everyone wrote a word on the face小题4: At last, __________.

A.both Jefferson and the man were safe to cross the river

B.only Jefferson crossed the river

C.only the man crossed the river

D.Jefferson’s horse died小题5:From the passage, we know that _________.

A.all the travelers were kind

B.Jefferson was helpful

C.nobody knew the president

D.the man gave Jefferson some money