一、选择题

　1.已知抛物线y2=2px(p>0)的焦点为F，点P1(x1，y1)，P2(x2，y2)，P3(x3，y3)在抛物线上，且2x2=x1+x3，则有()

　A.|FP1|+|FP2|=|FP3|

　B.|FP1|2+|FP2|2=|FP3|2

　C.2|FP2|=|FP1|+|FP3|

　D.|FP2|2=|FP1|·|FP3|

　答案：C　解题思路：抛物线的准线方程为x=-，由定义得|FP1|=x1+，|FP2|=x2+，|FP3|=x3+，则|FP1|+|FP3|=x1++x3+=x1+x3+p，2|FP2|=2x2+p，由2x2=x1+x3，得2|FP2|=|FP1|+|FP3|，故选C.

　2.与抛物线y2=8x相切倾斜角为135°的直线l与x轴和y轴的交点分别是A和B，那么过A，B两点的最小圆截抛物线y2=8x的准线所得的弦长为()

　A.4B.2　C.2D.

　答案：C　命题立意：本题考查直线与抛物线及圆的位置关系的应用，难度中等.

　解题思路：设直线l的方程为y=-x+b，联立直线与抛物线方程，消元得y2+8y-8b=0，因为直线与抛物线相切，故Δ=82-4×(-8b)=0，解得b=-2，故直线l的方程为x+y+2=0，从而A(-2,0)，B(0，-2)，因此过A，B两点最小圆即为以AB为直径的圆，其方程为(x+1)2+(y+1)2=2，而抛物线y2=8x的准线方程为x=-2，此时圆心(-1，-1)到准线的距离为1，故所截弦长为2=2.

　3.如图，过抛物线y2=2px(p>0)的焦点F的直线l交抛物线于点A，B，交其准线于点C，若|BC|=2|BF|，且|AF|=3，则此抛物线的方程为()

　A.y2=9x　B.y2=6x

　C.y2=3x　D.y2=x

　答案：C　命题立意：本题考查抛物线定义的应用及抛物线方程的求解，难度中等.

　解题思路：如图，分别过点A，B作抛物线准线的垂线，垂足分别为E，D，由抛物线定义可知|AE|=|AF|=3，|BC|=2|BF|=2|BD|，在RtBDC中，可知BCD=30°，故在RtACE中，可得|AC|=2|AE|=6，故|CF|=3，则GF即为ACE的中位线，故|GF|=p==，因此抛物线方程为y2=2px=3x.

　4.焦点在x轴上的双曲线C的左焦点为F，右顶点为A，若线段FA的中垂线与双曲线C有公共点，则双曲线C的离心率的取值范围是()

　A.(1,3) B.(1,3]

　C.(3，+∞) D.[3，+∞)

　答案：D　命题立意：本题主要考查双曲线的离心率问题，考查考生的化归与转化能力.

　解题思路：设AF的中点C(xC,0)，由题意xC≤-a，即≤-a，解得e=≥3，故选D.

　5.过点(，0)引直线l与曲线y=相交于A，B两点，O为坐标原点，当AOB的面积取值时，直线l的斜率等于()

　A. B.- C.± D.-

　答案：B　命题透析：本题考查直线与圆的位置关系以及数形结合的数学思想.

　思路点拨：由y=，得x2+y2=1(y≥0)，即该曲线表示圆心在原点，半径为1的上半圆，如图所示.

　故SAOB=|OA||OB|·sin AOB=sin AOB，所以当sin AOB=1，即OAOB时，SAOB取得值，此时O到直线l的距离d=|OA|sin 45°=.设此时直线l的方程为y=k(x-)，即kx-y-k=0，则有=，解得k=±，由图可知直线l的倾斜角为钝角，故k=-.

　6.点P在直线l：y=x-1上，若存在过P的直线交抛物线y=x2于A，B两点，且|PA|=|AB|，则称点P为“正点”，那么下列结论中正确的是()

　A.直线l上的所有点都是“正点”

　B.直线l上仅有有限个点是“正点”

　C.直线l上的所有点都不是“正点”

　D.直线l上有无穷多个点(点不是所有的点)是“正点”

　答案：A　解题思路：本题考查直线与抛物线的定义.设A(m，n)，P(x，x-1)，则B(2m-x,2n-x+1)， A，B在y=x2上， n=m2,2n-x+1=(2m-x)2，消去n，整理得关于x的方程x2-(4m-1)x+2m2-1=0， Δ=8m2-8m+5>0恒成立， 方程恒有实数解.

　二、填空题

　7.设A，B为双曲线-=1(b>a>0)上两点，O为坐标原点.若OAOB，则AOB面积的最小值为________.

　答案：　解题思路：设直线OA的方程为y=kx，则直线OB的方程为y=-x，则点A(x1，y1)满足故x=，y=，

　|OA|2=x+y=;

　同理|OB|2=.

　故|OA|2·|OB|2=·=.

　=≤(当且仅当k=±1时，取等号)， |OA|2·|OB|2≥，

　又b>a>0，

　故SAOB=|OA|·|OB|的最小值为.

　8.已知直线y=x与双曲线-=1交于A，B两点，P为双曲线上不同于A，B的点，当直线PA，PB的斜率kPA，kPB存在时，kPA·kPB=________.

　答案：　解题思路：设点A(x1，y1)，B(x2，y2)，P(x0，y0)，则由得y2=，y1+y2=0，y1y2=-，

　x1+x2=0，x1x2=-4×.

　由kPA·kPB=·====知kPA·kPB为定值.

　9.设平面区域D是由双曲线y2-=1的两条渐近线和抛物线y2=-8x的准线所围成的三角形(含边界与内部).若点(x，y)D，则目标函数z=x+y的值为______.

　答案：

　3　解题思路：本题考查双曲线、抛物线的性质以及线性规划.双曲线y2-=1的两条渐近线为y=±x，抛物线y2=-8x的准线为x=2，当直线y=-x+z过点A(2,1)时，zmax=3.

　三、解答题

　10.已知抛物线y2=4x，过点M(0,2)的直线与抛物线交于A，B两点，且直线与x轴交于点C.

　(1)求证：|MA|，|MC|，|MB|成等比数列;

　(2)设=α，=β，试问α+β是否为定值，若是，求出此定值;若不是，请说明理由.

　解析：(1)证明：设直线的方程为：y=kx+2(k≠0)，

　联立方程可得得

　k2x2+(4k-4)x+4=0.

　设A(x1，y1)，B(x2，y2)，C，

　则x1+x2=-，x1x2=，

　|MA|·|MB|=|x1-0|·|x2-0|=，

　而|MC|2=2=，

　|MC|2=|MA|·|MB|≠0，

　即|MA|，|MC|，|MB|成等比数列.

　(2)由=α，=β，得

　(x1，y1-2)=α，

　(x2，y2-2)=β，

　即得：α=，β=，

　则α+β=，

　由(1)中代入得α+β=-1，

　故α+β为定值且定值为-1.

　11.如图，在平面直角坐标系xOy中，设点F(0，p)(p>0)，直线l：y=-p，点P在直线l上移动，R是线段PF与x轴的交点，过R，P分别作直线l1，l2，使l1PF，l2l，l1∩l2=Q.

　(1)求动点Q的轨迹C的方程;

　(2)在直线l上任取一点M作曲线C的两条切线，设切点为A，B，求证：直线AB恒过一定点;

　(3)对(2)求证：当直线MA，MF，MB的斜率存在时，直线MA，MF，MB的斜率的倒数成等差数列.

　解题思路：本题考查轨迹方程的求法及直线与抛物线的位置关系.(1)利用抛物线的定义即可求出抛物线的标准方程;(2)利用导数及方程根的思想得出两切点的直线方程，进一步求出直线恒过的定点;(3)分别利用坐标表示三条直线的斜率，从而化简证明即可.

　解析：(1)依题意知，点R是线段PF的中点，且RQ⊥FP，

　RQ是线段FP的垂直平分线. |QP|=|QF|.故动点Q的轨迹C是以F为焦点，l为准线的抛物线，其方程为：x2=4py(p>0).

　(2)设M(m，-p)，两切点为A(x1，y1)，B(x2，y2).

　由x2=4py得y=x2，求导得y′=x.

　两条切线方程为y-y1=x1(x-x1)，

　y-y2=x2(x-x2)，

　对于方程，代入点M(m，-p)得，

　-p-y1=x1(m-x1)，又y1=x，

　-p-x=x1(m-x1)，

　整理得x-2mx1-4p2=0.

　同理对方程有x-2mx2-4p2=0，

　即x1，x2为方程x2-2mx-4p2=0的两根.

　x1+x2=2m，x1x2=-4p2.

　设直线AB的斜率为k，k===(x1+x2)，

　所以直线的方程为y-=(x1+x2)(x-x1)，展开得：

　y=(x1+x2)x-，

　将代入得：y=x+p.

　直线恒过定点(0，p).

陕西2017年高考英语阅读理解训练题及答案

　30. Why do more choices of goods give rise to anxiety?

　A. Professionals find it hard to decide on a suitable product.

　B. People are likely to find themselves overcome by business persuasion.

　C. Shoppers may find themselves lost in the broad range of items.

　D. Companies and advertisers are often misleading about the range of choice.

　31. By using computers as an example, the author wants to prove that .

　A. advanced products meet the needs of people

　B. products of the latest design fold the market

　C. competitions are fierce in high-tech industry

　D. everyday goods need to be replaced often

　32. What is this passage mainly about?

　A. The variety of choices in modern society.

　B. The opinions on people?s right in different countries.

　C. The problems about the availability of everyday goods.

　D. The helplessness in purchasing decisions.

　D

　Mr William Shakespeare and the Internet

　Explanation of Contents

　This is the fourth edition of these pages. It is hard to believe, but once again they are new and improved. My motive in publishing these pages remains to help and stimulate others in Shakespeare studies, and especially those who might contribute their work to the Internet. The spirit of altruism (利他主义) that originally built the Internet is not quite gone, though, sadly, through the pressure of time and profit has lessened.

　A major new addition to the pages is a Shakespeare Timeline, which is an online biography mounted at this site. The problems with searching for Shakespeare resources using the available Search Engines are:

　---- It is difficult to focus most searches so that you get a manageable number of relevant hits;

　---- It is impossible by simply reading an abstract(摘要) to make any distinction between the output of a Junior High School student and that of a professional researcher.

　Another change in these pages over previous editions is the ?What?s News? page. If you come away from these pages with the feeling that they are very useful but slightly pedantic (学究的), I will have realized my goal.

　An Apology

　I am continually apologizing to the many who have written me requesting revisions of the pages. We are all too busy. I simply have not had the time to dedicate to these pages that I wish. But I love the material and so have, at long last, made some time to update them.

　A Reminder to Young Students

　These pages contain the best links I can find to Shakespeare on the Internet. As a reminder, I would say I very much enjoy hearing from people who view and use these pages. If you want to do Shakespeare research using the web, this page is a great starting point, and I keep it as current as I can. The web is in its infancy(初期in bringing good, scholarly content to students. Don?t forget the best, if not quickest, resources are still in your library.

　33. The passage is written to ________.

　A. introduce the fourth edition of these pages

　B. make an apology to readers

　C. show off these pages to readers

　D. let Shakespeare researchers buy these pages

　34. When searching for Shakespeare resources using Search Engines, you ________.

　A. can easily recognize what the abstract means

　B. will waste some time in finding what you want

　C. will often come into the ?What?s News? pages

　D. will find something special on your computers

　35. Which of the following can best conclude the last paragraph?

　A. The writer will often read letters from those who use these pages.

　B. The writer of the passage is very selfish.

　C. The web was just created four years ago.

　D. Shakespeare researchers should first of all refer to these pages.

　第二节 根据短文内容，从短文后的选项中选出能填入空白处的最佳选项。选项中有两项为多余选项。

　As a teen, you?re going through big changes physically and mentally. Your interests are increasing. 36 . Here is the challenge: Kids need to explore the world in new ways, and parents need to protect them from the dangers that are all out in that world. These conflicts can easily set off fireworks in otherwise calm houses. Sometimes conflicts can?t be avoided. But by paying attention to the building blocks of successful relationships, you can work towards making home a happy and healthy place for you and your parents.

　For instance, try to find a time to talk when your parents are not angry, tired, distracted or hungry. A good time to talk is when you?re all relaxed. Timing is everything. If the conversation begins to turn into an argument, you?d better calmly and coolly ask to stop the conversation for now. 37 . Listen to what your parents are saying, and repeat it back to them. This shows them that you?re listening. 38 . Respect is the building block of good communication. People who respect each other and care about each others? feelings can disagree without getting things ugly. 39 . How do you build trust? Trust comes by actually doing what you say you?re going to do. Some teens find that doing fun activities with their parents can improve their relationships. Sometimes we forget that parents are more than rule-maker?they?re interestingpeople who like to watch movies and go shopping?just like their teenagers!

　What do you do if you are trying your best, but your relationship with your parents continues to be rocky? 40 You can find supportive adults, such as a teacher or a coach, who can lend an ear. Remember you can only change your own behavior. Your parents are the only ones who can change theirs.

　A. It also gives them a chance to clear things up if you?re not on the same page.

　B. You can pick it up again when everyone?s more relaxed.

　C. And then you?ll be able to accept what your parents say.

　D. Faced with the challenge, children don?t know what to do

　E. You are more likely to get along with your parents and have more independence if

　your parents believe in you.

　F. And your desire to take control of your own life is growing.

　G. You may consider seeking outside help.

参考答案：

　21-24 DBAC 25? 28 BDDC 29- 32 .BCBD 33-35 ABD 36-40 FBAEG

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