找最短距离已知：直线L两侧有a,b两点.直线上取一点使2a+b最短请写出证明过程a和b代表点a和点b所所对应的线段

题目详情

找最短距离
已知：直线L两侧有a,b两点.直线上取一点使2a+b最短
请写出证明过程 a和b代表点a和点b所所对应的线段

▼优质解答

答案和解析

2a+b=2(h^2+x^2)^(1/2)+(H^2+(m-x)^2)^(1/2)=f(x)
f(x)'=-(m-x)/[根号(H^2+(m-x)^2)]+(2x)/[根号(h^2+x^2)]
f(x)'=0
(m-x)/[根号(H^2+(m-x)^2)]=(2x)/[根号(h^2+x^2)]
(m-x)^2/(H^2+(m-x)^2)=(4x^2)/(h^2+x^2)
(h^2)(m^2)-2(h^2)mx+(h^2 -4H^2-3m^2)x^2+6mx^3-3x^4=0
这是1元4次方程.
四个根如下.
{x -> m/2 -
1/2 ,(m^2 + 1/3 (h^2 - 4 H^2 - 3 m^2) +
1/9 (-h^2 + 4 H^2 + 3 m^2) + (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) +
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3))) -
1/2 ,(2 m^2 +
4/9 (h^2 - 4 H^2 - 3 m^2) - (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) -
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(
1/3)) - (-((16 h^2 m)/3) + 8 m^3 -
8/
3 m (-h^2 + 4 H^2 + 3 m^2))/(4 ,(m^2 + 1/3 (h^2 - 4 H^2 - 3 m^2) +
1/
9 (-h^2 + 4 H^2 + 3 m^2) + (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) -
2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 +
972 h^2 m^4 + 324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) +
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) -
2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)))))},
{x ->
m/2 - 1/
2 ,(m^2 + 1/3 (h^2 - 4 H^2 - 3 m^2) +
1/9 (-h^2 + 4 H^2 + 3 m^2) + (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) +
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3))) +
1/2 ,(2 m^2 +
4/9 (h^2 - 4 H^2 - 3 m^2) - (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) -
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(
1/3)) - (-((16 h^2 m)/3) + 8 m^3 -
8/
3 m (-h^2 + 4 H^2 + 3 m^2))/(4 ,(m^2 + 1/3 (h^2 - 4 H^2 - 3 m^2) +
1/
9 (-h^2 + 4 H^2 + 3 m^2) + (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) -
2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 +
972 h^2 m^4 + 324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) +
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) -
2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)))))},
{x ->
m/2 + 1/
2 ,(m^2 + 1/3 (h^2 - 4 H^2 - 3 m^2) +
1/9 (-h^2 + 4 H^2 + 3 m^2) + (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) +
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3))) -
1/2 ,(2 m^2 +
4/9 (h^2 - 4 H^2 - 3 m^2) - (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) -
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(
1/3)) + (-((16 h^2 m)/3) + 8 m^3 -
8/
3 m (-h^2 + 4 H^2 + 3 m^2))/(4 ,(m^2 + 1/3 (h^2 - 4 H^2 - 3 m^2) +
1/
9 (-h^2 + 4 H^2 + 3 m^2) + (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) -
2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 +
972 h^2 m^4 + 324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) +
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) -
2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)))))},
{x ->
m/2 + 1/
2 ,(m^2 + 1/3 (h^2 - 4 H^2 - 3 m^2) +
1/9 (-h^2 + 4 H^2 + 3 m^2) + (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) +
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3))) +
1/2 ,(2 m^2 +
4/9 (h^2 - 4 H^2 - 3 m^2) - (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) -
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) - 2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(
1/3)) + (-((16 h^2 m)/3) + 8 m^3 -
8/
3 m (-h^2 + 4 H^2 + 3 m^2))/(4 ,(m^2 + 1/3 (h^2 - 4 H^2 - 3 m^2) +
1/
9 (-h^2 + 4 H^2 + 3 m^2) + (2^(
1/
3) (h^2 - 4 H^2 - 3 m^2)^2)/(9 (324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) -
2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 +
972 h^2 m^4 + 324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)) +
1/(9 2^(1/3))((324 h^4 m^2 - 972 h^2 m^4 -
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) -
2 (h^2 - 4 H^2 - 3 m^2)^3 -
,(-4 (h^2 - 4 H^2 - 3 m^2)^6 + (-324 h^4 m^2 + 972 h^2 m^4 +
324 h^2 m^2 (h^2 - 4 H^2 - 3 m^2) +
2 (h^2 - 4 H^2 - 3 m^2)^3)^2))^(1/3)))))}
其中有个根在0和m之间,带入原式即可.

看了找最短距离已知：直线L两侧有...的网友还看了以下：

读读下面句子,说说怎样才能真正成为学习的主人.（1）学问学问,既要学又要问.学与问是相辅相成的,只有　2020-03-30 …

为什么说“在求知的过程中,我们还要善于把勤学好问和观察思考结合起来“?本句话摘自苏教版小学语文六年　2020-05-23 …

读读下面的句子,说说怎样才能真正成为学习的主人.（1)学问学问,既要学又要问.学与问是相辅相成的, 　2020-05-23 …

在求知的过程中,我们还要善于把勤学好问和观察思考结合起来.举事例　2020-06-23 …

点斜式方程求解答!y-a=k（x-b）已知的是应该带到ab的位置,另一已知点带到xy位置么? 　2020-08-01 …

阅读短文，完成练习。在求知的过程中，我们要善于把勤学好问和观察思考结合起来。北宋有个大科学家，名叫　2020-08-03 …

阅读理解。学与问（节选）在求知的过程中，我们还要善于把勤学好问和观察思考结合起来。北宋有个大科学家，　2020-11-04 …

挫折是每个人都无法避免的，受挫折的过程往往就是获得真知的过程。走过崎岖小路，才能真正体味生活的欢乐；　2020-11-16 …

阅读理解。学与问（节选）在求知的过程中，我们还要善于把勤学好问和观察思考结合起�阅读理解。学与问（节　2020-11-27 …

论语中说明从旧知中不断获得新知的过程是哪三句话　2020-11-27 …