高中数学训练题及解析——直线与圆的位置关系.doc

平面解析几何——直线与圆的位置关系

一、选择题
1．已知直线l过点(－2,0)，当直线l与圆x2＋y2＝2x有两个交点时，其斜率k的取值范围是(　　)
A．(－2，2)　　　　　B．(－，)
C．(－，) D．(－，)
答案　C
解析　设l的方程y＝k(x＋2)，即kx－y＋2k＝0.
圆心为(1,0)．由已知有<1，∴－<k<.
2．直线xsinθ＋ycosθ＝2＋sinθ与圆(x－1)2＋y2＝4的位置关系是(　　)
A．相离 B．相切
C．相交 D．以上都有可能
答案　B
解析　圆心到直线的距离d＝＝2.
所以直线与圆相切．
3．平移直线x－y＋1＝0使其与圆(x－2)2＋(y－1)2＝1相切，则平移的最短距离为(　　)
A.－1 B．2－
C. D.－1与＋1

答案　A
解析　如图，圆心(2,1)到直线l0：x－y＋1＝0的距离d＝＝，圆的半径为1，故直线l0与l1的距离为－1，
∴平移的最短距离为－1，故选A.
4．已知圆O1：(x－a)2＋(y－b)2＝4；O2：(x－a－1)2＋(y－b－2)2＝1(a，b∈R)，那么两圆的位置关系是(　　)
A．内含 B．内切
C．相交 D．外切
答案　C
解析　由两圆方程易知其圆心坐标分别为O1(a，b)、O2(a＋1，b＋2)，经计算得：O1O2＝，由于R－r＝1<O1O2＝<R＋r＝3，故两圆相交．
5．函数y＝f(x)的图象是圆心在原点的单位圆在Ⅰ、Ⅲ象限内的两段圆孤，如图，则不等式f(x)<f(－x)＋2x的解集为(　　)

A．(－1，－)∪(0，)
B．(－1，－)∪(，1)
C．(－，0)∪(0，)
D．(－，0)∪(，1)
答案　D
6．由直线y＝x＋1上的一点向圆(x－3)2＋y2＝1引切线，则切线长的最小值为(　　)
A．1 B．2
C. D．3
答案　C
解析　

设直线上一点P，切点为Q，圆心为M，
则|PQ|即为切线长，MQ为圆M的半径，长度为1，
|PQ|＝＝，要使|PQ|最小，即求|PM|最小，此题转化为求直线y＝x＋1上的点到圆心M的最小距离，设圆心到直线y＝x＋1的距离为d，则d＝＝2，
∴|PM|最小值为2，|PQ|＝＝＝，选C.
7．若圆O1方程为：(x＋1)2＋(y＋1)2－4＝0，圆O2方程为：(x－3)2＋(y－2)2－1＝0，则方程(x＋1)2＋(y＋1)2－4＝(x－3)2＋(y－2)2－1表示的轨迹是(　　)
A．线段O1O2的中垂线
B．过两圆的公切线交点且垂直于线段O1O2的直线
C．两圆公共弦所在的直线
D．一条直线且该直线上的点到两圆的切线长相等
答案　D
解析　∵圆心距|O1O2|＝＝5>2＋1＝3，∴两圆相离．
把所给的轨迹方程化简得4x＋3y－7＝0
显然线段O1O2的中点不在直线4x＋3y－7＝0上，排除A、C，由计算知，到两圆的切线长相等的点的轨迹恰为直线4x＋3y－7＝0.
8．已知圆C：x2＋y2＝1，点A(－2,0)及点B(2，a)，从A点观察B点，要使视线不被圆C挡住，则a的取值范围是(　　)
A．(－∞，－1)∪(－1，＋∞)
B．(－∞，－2)∪(2，＋∞)
C．(－∞，－)∪(，＋∞)
D．(－∞，－4)∪(4，＋∞)
答案　C
解析　解法一：(直接法)写出直线方程，将直线与圆相切转化为点到直线的距离来解决．
过A、B两点的直线方程为y＝x＋，
即ax－4y＋2a＝0，
则d＝＝1，化简后，得3a2＝16，解得a＝±.再进一步判断便可得到正确答案为C.
解法二：

设AB1直线方程为⇒(1＋k2)x2＋4k2x＋4k2－1＝0，Δ＝0，k＝±，
直线AB1方程为y＝(x＋2)，直线AB2方程为y＝－(x＋2)，可得B1(2，)，B2(2，－)，要使从A看B不被圆挡住，B纵坐标即实数a的取值范围为(－∞，－)∪(，＋∞)．
9．若圆(x－3)2＋(y＋5)2＝r2上有且只有两个点到直线4x－3y－2＝0的距离等于1，则半径r的取值范围是(　　)
A．(4,6) B．[4,6)
C．(4,6] D．[4,6]
答案　A
二、填空题
10．已知直线x＋y＝a与圆x2＋y2＝4交于A，B两点，且|＋|＝|－|(其中O为坐标原点)，则实数a等于________．
答案　±2
解析　由|＋|＝|－|知OA⊥OB，所以由题意可得＝，所以a＝±2.
11．过点M(1,2)的直线l将圆(x－2)2＋y2＝9分成两段弧，其中的劣弧最短时，直线l的方程为________．
答案　x－2y＋3＝0
解析　设圆心为N(2,0)，由圆的性质得直线l⊥MN时，形成的劣弧最短，由点斜式得直线l的方程为x－2y＋3＝0.
12．(2010·江西卷，理)直线y＝kx＋3与圆(x－3)2＋(y－2)2＝4相交于M，N两点，若|MN|≥2，则k的取值范围是________．
答案　[－，0]

解析　如图，记题中圆的圆心为C(3,2)，作CD⊥MN于D，则|CD|＝，于是有|MN|＝2|MD|＝2＝2≥2，即4－≥3，解得－≤k≤0.
13．若直线y＝x＋b与曲线x＝恰有一个公共点，则b取值范围是__________．
答案　－1＜b≤1或b＝－

解析　x＝⇔x2＋y2＝1(x≥0)
方程x2＋y2＝1(x≥0)所表示的曲线为半圆(如图)
当直线与圆相切时或在l2与l3之间时，适合题意．
三、解答题
14．已知圆C：x2＋y2＋2x－4y＋3＝0.若圆C的切线在x轴和y轴上的截距的绝对值相等，求此切线的方程．
解析　∵切线在两坐标轴上截距的绝对值相等，
∴切线的斜率是±1.
设切线方程为y＝－x＋b或y＝x＋c，分别代入圆C的方程得2x2－2(b－3)x＋(b2－4b＋3)＝0
或2x2＋2(c－1)x＋(c2－4c＋3)＝0，
由于相切，则方程有等根，
即b＝3或b＝－1，c＝5或c＝1.
故所求切线方程为：
x＋y－3＝0，x＋y＋1＝0，x－y＋5＝0，x－y＋1＝0.
15．(2011·北京海淀区期末)已知圆C经过点A(－2,0)，B(0,2)，且圆心C在直线y＝x上，又直线l：y＝kx＋1与圆C相交于P、Q两点．
(1)求圆C的方程；
(2)若·＝－2，求实数k的值；
(3)过点(0,1)作直线l1与l垂直，且直线l1与圆C交于M、N两点，求四边形PMQN面积的最大值．
解　设圆心C(a，a)，半径为r.因为圆C经过点A(－2,0)，B(0,2)，
所以|AC|＝|BC|＝r，易得a＝0，r＝2，
所以圆C的方程是x2＋y2＝4.
(2)因为·＝2×2×cos〈，〉＝－2，且与的夹角为∠POQ，
所以cos∠POQ＝－，∠POQ＝120°，
所以圆心到直线l：kx－y＋1＝0的距离d＝1，
又d＝，所以k＝0.
(3)设圆心O到直线l，l1的距离分别为d，d1，四边形PMQN的面积为S.
因为直线l，l1都经过点(0,1)，且l⊥l1，
根据勾股定理，有d＋d2＝1.
又易知|PQ|＝2×，|MN|＝2×，
所以S＝·|PQ|·|MN|，即
S＝×2××2×＝2＝2≤2 ＝2 ＝7，
当且仅当d1＝d时，等号成立，所以S的最大值为7.

技术官员村位于位于亚运城东部，主干道二以东石楼涌以西的地块，占地面积、m2，总建筑面积、m2，共包括地下室南区、地下室北区、地上部分1栋～12栋、服务中心、室外工程等多个单体工程。其中住宅面积m2，共12栋，1～7栋建筑层数为11层，8～12栋11层（局部复式顶层），首层局部架空，布置公建配套设施。integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed,