题目
题型:不详难度:来源:
两条互相平行的直线分别过点A(6,2)和B(-3,-1),并且各自绕着A,B旋转,如果两条平行直线间的距离为d.
求:1)d的变化范围;
2)当d取最大值时两条直线的方程.
答案
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153419-60726.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153419-81974.png)
(2)由于当d最大时,AB与直线垂直,所以可以利用AB的斜率求出直线的斜率,进而求出其直线方程.
(1)方法一:①当两条直线的斜率不存在时,即两直线分别为x=6和x=-3,则它们之间的距离为9. ………………2分
②当两条直线的斜率存在时,设这两条直线方程为
l1:y-2=k(x-6),l2:y+1=k(x+3),
即l1:kx-y-6k+2=0,l2:kx-y+3k-1=0, ………………4分
∴d==. ………………6分
即(81-d2)k2-54k+9-d2=0. ………………8分
∵k∈R,且d≠9,d>0,
∴Δ=(-54)2-4(81-d2)(9-d2)≥0,即0<d≤3且d≠9.………………12分
综合①②可知,所求d的变化范围为(0,3].
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153420-89317.jpg)
方法二:如图所示,显然有0<d≤|AB|.
而|AB|==3.
故所求的d的变化范围为(0,3].
(2)由图可知,当d取最大值时,两直线垂直于AB.
而kAB==,
∴所求直线的斜率为-3. 故所求的直线方程分别为
y-2=-3(x-6),y+1=-3(x+3),即3x+y-20=0和3x+y+10=0.
核心考点
试题【(本题满分12分)两条互相平行的直线分别过点A(6,2)和B(-3,-1),并且各自绕着A,B旋转,如果两条平行直线间的距离为d.求:1)d的变化范围;2)当d】;主要考察你对直线方程的几种形式等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-68793.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-97340.png)
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![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-92972.png)
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![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-90164.jpg)
(I)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-92972.png)
(II)设点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-81942.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-92972.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-61008.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-50642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-92972.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-49093.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-82612.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-49093.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-82612.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153404-81942.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153405-67076.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153405-89280.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153405-89457.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153405-43606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153358-15345.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153358-41556.png)
(Ⅰ)当直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153358-65732.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153358-84420.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153358-65732.png)
(Ⅱ)当直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153358-65732.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153358-65732.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153358-65732.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153351-65267.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153352-38410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153352-40379.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022153340-66085.png)
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A.![]() | B.![]() | C.![]() | D.![]() |
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