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118000300@qq.com:人教A版数学必修二4-2-1《直线与圆的位置关系》同步练习

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2014 高中数学 4-2-1 直线与圆的位置关系同步练习 新人教 A 版必 修2 一、选择题 1.过点(2,1)的直线中,被圆 x +y -2x+4y=0 截得的弦最长的直线 的方程是( A.3x-y-5=0 C.3x-y-1=0 [答案] A [解析] x +y -2x+4y=0 的圆心为(1,-2),截得弦最长的直线必过点(2,1)和圆心 (1,-2) ∴直线方程为 3x-y-5=0,故选 A. 2.若过点 A(4,0)的直线 l 与曲线(x-2) +y =1 有公共点,则直线 l 的斜率的取值范 围为( ) B.[- 3, 3] D.?- 2 2 2 2 2 2 ) B.3x+y-7=0 D.3x+y-5=0 A.(- 3, 3) C.?- ? ? 3 3? , ? 3 3? ? ? 3 3? , ? 3 3? [答案] D [解析] 解法 1:如图,BC=1,AC=2, ∴∠BAC=30°, ∴- 3 3 ≤k≤ . 3 3 解法 2:设直线 l 方程为 y=k(x-4),则由题意知, |2k-0-4k| 3 3 ≤1,∴- ≤k≤ . 2 3 3 1+k 解法 3:过 A(4,0)的直线 l 可设为 x=my+4,代入(x-2 ) +y =1 中得: (m +1)y +4my+3=0, 由 Δ =16m -12(m +1)=4m -12≥0 得 2 2 2 2 2 2 2 m≤- 3或 m≥ 3. 1 ? 3 ? ? 3? ∴l 的斜率 k= ∈?- ,0?∪?0, ?,特别地,当 k=0 时,显然有公共点, m ? 3 3 ? ? ? ∴k∈?- ? ? 3 3? , ?. 3 3? 3.已知直线 ax+by+c=0(abc≠0)与圆 x +y =1 相切,则三边长分别为|a|、|b|、 |c|的三角形( ) B.是直角三角形 D.不存在 2 2 A.是锐角三角形 C.是钝角三角形 [答案] B [解析] 由已知得: 故选 B. |c| a +b 2 2 =1,∴a +b =c 2 2 2 4.点 M 在圆(x-5) +(y-3) =9 上,点 M 到直线 3x+4y-2=0 的最短距离为( A.9 C.5 [答案] D B.8 D.2 2 2 ) |15+12-2| [解析] 由圆心到直线的距离 d= =5>3 知直线与圆相离,故最短距离为 d 2 2 3 +4 -r=5-3=2,故选 D. 5. 过点 A(11,2)作圆 x +y +2x-4y-164=0 的弦,其中弦长为整数的共有 ( A.16 条 C.32 条 [答案] C [解析] 圆 C:(x+1) +(y-2) =169 的圆心 C(-1,2),点 A(11,2)在⊙C 内,|AC| =12, ∴最短弦长为 2 169-12 =10,最长弦长为 26. ∴弦长为整数的弦共有(26-10)×2=32 条. [点评] 弦长为 10 和 26 的各一条,弦长为 11 至 25 的各有 2 条,计数时应区分. 6.从点 P(x,3)向圆(x +2) +(y+2) =1 引切线,切线长的最小值为( A.4 C.5 [答案] B [解析] 当 P(x,3)到圆心 C(-2, -2)距离最小时, 过 P 点向圆所作切线的切线长最小. |PC|= (x+2) +(3+2) = (x+2) +25 ∴|PC|min=5. ∴切线长的最小值为 5 -1=2 6. 故选 B. 2 2 2 2 2 2 2 2 2 2 2 ) B.17 条 D.34 条 ) B.2 6 D.5.5 [点评] 圆心 C(-2,-2),半径 r=1,设切点 M,则|PM| =|PC| -|CM| =|PC| -1, ∴|PM|= |PC| -1= (x+2) +(3+2) -1= (x+2) +24≥ 24=2 6, 一般地, 由圆外 一点 P(x0, y0)向圆(x-a) +(y-b) =r 所引切线长 l= (x0-a) +(y0-b) -r , 由圆外一 点 P(x0,y0),向圆 x +y +Dx+Ey+F=0 所引切线长 l= x0+y0+Dx0+Ey0+F. *7.已知点 M(a,b)(ab≠0)是圆 x +y =r 内一点,直线 m 是以 M 为中点的弦所在的直 线,直线 l 的方程是 ax+by=r ,则( A.l∥m 且 l 与圆相交 C.l∥m 且 l 与圆相离 [答案] C [解析] 直线 m 的方程为,ax+by-a -b =0, ∴l∥m,圆心到直线 ax+by=r 的距离 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ) B.l⊥m 且 l 与圆相切 D.l⊥m 且 l 与圆相离 d= |r | 2 a2+b2 >r, ∴直线 l 与圆相离,故选 C. 8.曲线 y=1+ 4-x (-2≤x≤2)与直线 y=k(x-2)+4 有两个交点时,实数 k 的取 值范围是( A.? ) B.? 2 ? 5 ,+∞? ? ?16 ? ? 5 ,3? ? ?12 4? 5? ? C.?0, ? ? 12? [答案] B [解析] 如图 ?1 3? D.? , ? ?3 4? 直线 y=k(x-2)+4 与半圆 y=1+ 4-x (-2≤x≤2)有两个交点,应满足 KMB<k≤KMA, 5 3 即 <k≤ 12 4 故选 B. 9.已知圆 x +y =4,过点 A(4,0)作圆的割线 ABC,则弦 BC 中点的轨迹方程为( 1? ? 2 2 A.(x-1) +y =4 ?-1≤x< ? 2? ? 2 2 2 ) B.(x-1) +y =4 (0≤x<1) 1? ? 2 2 C.(x-2) +y =4 ?-1≤x< ? 2 2 2 ? ? D.(x-2) +y =4 (0≤x<1) [答

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