问题 解答题

对称轴为坐标轴,顶点在坐标原点的抛物线C经过两点A(a,2a)、B(4a,4a)(其中a为正常数),

(1)求抛物线C的方程;

(2)设动点T(m,0)(m>a),直线AT、BT与抛物线C的另一个交点分别为A1、B1,当m变化时,记所有直线A1B1组成的集合为M,求证:集合M中的任意两条直线都相交且交点都不在坐标轴上。

答案

解:(1)当抛物线焦点在x轴上时,设抛物线方程y2=2px,

∴p=2a,

∴y2=4ax;

当抛物线焦点在y轴上时,设抛物线方程x2=2py,

∴方程无解,

∴抛物线不存在。

(2)设A1(as2,2as)、B1(at2,2at),T(m,0)(m>a),

∴as2+(m-a)s-m=0,

∵(as+m)(s-1)=0,

∴A1,-2m),

∵2at2+(m-4a)t-2m=0,

∴(2at+m)(t-2)=0,

∴t=

∴B1,-m),

的直线方程为y+2m=

∵直线的斜率为在(a,+∞)单调,

∴所以集合M中的直线必定相交,

∵直线的横截距为在(a,+∞)单调,纵截距为在(a,+∞)单调,

∴任意两条直线都相交且交点都不在坐标轴上。

阅读理解

Ever tried taking smart drugs to perform better in exams? They certainly work. Before last summer's exam, a few of my mates, merely out of curiosity, took a considerable amount of Modafinil, a drug normally used for narcolepsy(嗜眠发作症). Guess what? While I was feeling sleepy, bored by revision, my mates were more focused.

Modafinil is a smart drug that gives a user a feeling of staying awake for hours. It also sharpens the mind, improves memory and helps problem-solving.

But it does more than just keeping you awake. It disturbs your mental system, making you desperate to do what you are doing. You just don't want to do anything else but revise all the time non-stop.

A recent study discovers that healthy people use smart drugs, like Modafinil, to get down to and complete tasks they have been putting off, because these tasks seem more enjoyable when taking these drugs. This might sound like what a stressed student desires. However, the pills have a range of side-effects.

"At present no evidence shows that these drugs are safe in healthy people," as a medical expert puts it, "Ordering online, though easy, is a dangerous way to obtain drugs. You never know what you're actually purchasing. "

In my experience, Modafinil changes people's behaviour too. Over those weeks my friends became different people—in turn aggressive, cold. Even eating became "a waste of time" and so did conversation.

A BBC survey found that of those who had tried smart drugs before, 92% would do so again. My friends say they'd happily do so without considering the practice a form of cheating.

I admit I was curious—but not enough to try it. Seeing the strange behaviour of other users, I feel a little bit upset and scary. Modafinil may promise to change your grades, but it might also change the way you act. Don't say you haven't been warned!

小题1:According to the author, his mates took Modafinil for the purpose of______.

A.escaping from reality

B.getting better exam results

C.curing their narcolepsy

D.leading an active life小题2:Taking the smart drug can lead to ______.

A.cheating in the exam

B.dropping out of school

C.youth crime(犯罪)

D.drug addiction(上瘾)小题3:What does the author think of his mates' taking the smart drugs?

A.Worrying.

B.Inspiring.

C.Acceptable.

D.Reasonable.小题4:Which statement about Modafinil is wrong?

A.Modafinil sharpens the mind and helps problem-solving.

B.Modafinil more than just keeps you awake.

C.Modafinil has a range of side effects.

D.Modafinil is safe in healthy people.小题5:Who are the target readers of this passage?

A.Teachers.

B.Doctors.

C.Scientists.

D.Students.

单项选择题