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.

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