椭圆中几个新发现的性质

性质1 已知椭圆(x2)/(a2) (y2)/(b2)=1(a＞0,b＞0)(包括圆在内)上有一点P,过P分别引直线y=(b)/(a)x及y=-(b)/(a)x的平行线,分别交x轴于M,N,交y轴于R,Q,O为原点,则:     

再探椭圆“切准点”对焦点的若干性质

文[1]定义了椭圆的切准点:椭圆(x~2)/(a~2)+(y~2)/(b~2)=1(a>b>0)上点M(x_0,y_0)(除长轴两顶点)处的切线l交右准线l_2:x=(a~2)/c于P,交左准线l_1:x=-(a~2)/c于Q,则点P,Q为椭圆的切准点.笔者     

一题多变 触类旁通

题目已知:如图1,AM是△ABC中BC边上的中线,P是AM上任意一点,过点P作DE∥BC,交AB、AC分别于D、E. 求证:PD=PE. 证明:∵DE∥BC, ∴(PD)/(BM)=(AP)/(AM),(PE)/(MC)=(AP)/(AM),∴ (PD)/(BM)=(PE)/(MC), ∵BM=MC,∴PD=PE. 变式一已知:如图2,AM是△ABC中BC边上的中线,P是AM上     

圆锥曲线“中点弦”的斜率公式

设A(x1,y1),B(x2,y2)是圆锥曲线上不同的两点,G(xA,yB)是线段AB的中点,kAB是AB弦所在直线的斜率.则有:(1)椭圆(x2)/(a2)+(y2)/(b2)=1,kAB=-(b2xA)/(a2yB)(2)双曲线三(x2)/(a2)-(y2)/(b2)=1,kAB=-(b2xA)/(a2yB);(3)抛物线y2=2px(p>0),kAB=P/(yA).证明:(1)因A、B两点在椭圆(x2)/(a2)+(y2/b2)=1上,所以有     

系列讲座之十三——圆锥曲线(3)

本讲主要涉及向量与圆锥曲线之间的关系的一类竞赛问题. 例1 已知椭圆T:(x2)/(a2)+(y2)/(b2)＝1(a＞b＞0)和双曲线S:(x2)/(m2)+(y2)/(n2)＝1(m＞0, n＞0)具有相同的焦点F(2,0).设双曲线S经过第一象限的渐近线为l.若焦点F和椭圆T上方的顶点B关于l的对称点都在双曲线S上,求椭圆T和双曲线S的方程.     

双曲线渐近线方程统一形式的妙用

<正>我们知道,双曲线(x~2)/(a~2)-(y~2)/(b~2)=1的渐近线方程为y=±(b/a)x.一般地,还有下面的一些结论:(1)双曲线(x~2)/(a~2)-(y~2)/(b~2)=λ(λ>0)的渐近线方程亦为y=±bax,即xa±yb=0,就是(x~2)/(a~2)-(y~2)/(b~2)=0.(2)双曲线(x~2)/(a~2)-(y~2)/(b~2)=λ(λ<0)的渐近线方程亦为(x~2)/(a~2)-(y~2)/(b~2)=0,故双曲线(x~2)/(a~2)-(y~2)/(b~2)=λ(λ≠0)的渐近线方程为     

例谈高中数学解题中的检验问题

<正>一、解分式方程例1在实数范围内解关于x的方程(x²+x-2)/(x-1)=0。解:因为(x2+x-2)/(x-1)=0。解:因为(x²+x-2)/(x-1)=0,所以x2+x-2)/(x-1)=0,所以x²+x-2=0,则x=1或x=-2。检验:x=1时,x-1=0,舍去,则x=-2。点评:之所以要检验,是因为在解分式方程时把分式等于零转化成了分子等于零,这是一个不等价转换,从逻辑上说后者是前者的必要条件,满足分式方程的解必满足整式     

圆锥曲线的一个奇妙性质

笔者最近在研究圆锥曲线时,发现圆锥曲线的一个奇妙性质,现介绍如下:定理1已知椭圆E:(x~2)/(a~2) (y~2)/(b~2)=1(a>b>0),过不在椭圆E上的定点T(m,n)作定直线l:(mx)/(a~2) (ny)/(b~2)=1的垂线TD,垂足为D,过T引     

一脉相承 星火燎原——对两道姊妹题的探究

一、从联赛到自主招生,一脉相承题1(2010年全国高中数学联赛江西省预赛试题)已知椭圆(x~2)/(a~2)+(y~2)/(b~2)=1(a>b>0)和圆x~2+y~2=b~2,经过椭圆上的动点M作圆的两条切线,切点分别为P,Q,若直线PQ在x轴、y轴上的截距分别为m,n,证明:(a~2)/(n~2)+(b~2)/(m~2)=(a~2)/(b~2).题2(2014年华约试题)已知椭圆(x~2)/(a~2)+(y~2)/(b~2)=1(a>b>0)和圆x~2+y~2=b~2,经过椭圆上的动点M作圆的两条切线,切点分别为P,Q,直线PQ与坐标轴的交点分别为E,F,求AEOF面积的最小值.     

2004年全国新课程卷理科21题别解

题目 设双曲线C:(x2)/(a2)-y2=1(a>0)与直线l:x y=1相交于两个不同的点A、B. (Ⅰ)求双曲线C的离心率e的取值范围; (Ⅱ)设直线l与y轴的交点为P,且PA=(5)/(12)PB,求a的值.     

卡思卡特城堡

It is worthy of noting that, whilst Crookston Castle witnessed the earlier and happier portion of Mary's variegated life,     

不同的吃喝玩乐

一、吃和喝吃苹果 eat an apple, 吃药 take medicine,吃糖 have some sweets,吃饭 have one's meals,吃馆子 dine out,吃惊 be surprised/     

漏洞百出的教育

Many college freshmen arrive woefully unprepared to do college work, and as disadvantaged populations continue to grow, the share of the American work force that has made it through college is expected to plummet. Many experts blame that educational failure not just on high schools but also on colleges. School & College, a special report by The Chronicle, looks at efforts to fix the system. What reforms would better prepare students for college? What should schools and colleges be doing differently? How should state and federal officials help?     

只要努力,不要后悔

There are numbers of crossroads on our long and unpredictable life journey where we totally have no idea about which direction to choose. No matter what our decision is, we should not turn back, but face the music and go ahead instead. I am this kind of girl who always does try without regretting, one example is how I dealt with my love.     

浅谈移民问题

Migration occurs behind a variety of reasons and has a great effect on the whole world. People may migrate in order to improve their economic situation, or in order to escape civil strife, persecution, and environmental disasters. The impact of migration is complex, bringing both benefits anddisadvantages. This paper briefly talks about the causes of migration, the allocation of benefits, and the ways in which individual countries and the international community deal with this important subject.     

Sister Carrie, an Adherent of Desires

Sister Carrie is one of the most controversial characters in American literature.Thought as a "fallen woman" firstly,she was defined as a "new woman" by some critics later. However, by digging into the motivaton behind the whole process of Carrie's "success", the relationship between Carrie and her creator (the author), the social conditions of then American, it can be found that Carrie has never been free-standing on her thought and she has never found her real-sdf even after becoming a famous actress. In a society dominated by mass consumerism Carrie is only an adherent of her own desires. She also is a representative of all those country girls flooded into cities, a symbol and a sacrifice of the urbanization of America in a time countryside was overcome by cities.     

Improvement of one-cycle controller by use of proportional integral differential controller

1.IntroductionOne-cyclecontrolmethod,whichwasproposedaboutonedecadeago[1],hasbecomeanattractivemethodinspecialfieldssuchaspowerfactorcorrection[2-6],switchingamplifiers[7,8],etc.Themainideaofthiscontrollerisbasedonintegrationofdiodevoltageinone-cycleandforcesittobeexactlyequaltothereferencevalue.Themainadvantageofthiscontrollerisitsrealtimeabilitytorejectthevariationofinputvoltage[1].Despitethisgreatability,ithasnogoodperformancesinrejectingofloaddisturbanceandfollowingreferencecommands.Espec…     

库克斯顿城堡

城堡, 是欧洲典型的建筑之一, 每个城堡都有一个美丽的传说或是一段历史。从本期开始, 我们将推出“城堡之旅”, 带各位读者去浏览、观赏欧洲各地的城堡, 开始一段文化与历史的旅途!     

胡主席在印度,我却在中国

Today, Sunday morning, I sit thinking and typing at my computer. My thoughts are in my homeland, India and my body is in China. The body is here and mind there for a good reason. Mr. Hu is president and I am resident and both are in each other's countries (what a difference the absence of one small letter makes!). Mr. Hu will return in a few days and I will stay on longer.