Find the length of the shortest path from the point (3,5) to the point (8,2) that touches the x-axis and also touches the y-axis

Question

My problem is that I can't create a line that passes through both points AND is the shortest path AND touches both axis lines.

score 4 · Answer 1 · answered Feb 10 '13 at 18:26

4

Hint: fix one point and look at paths from this point to a reflection of the other in both axes (you need to cross both) - eg from (3,5) to (-8,-2)

Think about how the proof by reflection works for one mirror, and apply it with two.

answered Feb 10 '13 at 18:26

Mark Bennet

100,194

score 3 · Answer 2 · answered Feb 10 '13 at 18:22

Hint: A path can be a sequence of segments. You can go from $(8,2)$ to $(0,y)$ to $(x,0)$ to $(3,5)$ Calculate the length of each segment, add them to get the total path length. Then you can differentiate to find the minimum in each variable-you will get two equations in two unknowns. Or, you can remember angle of incidence=angle of reflection and use that. I don't warrant that this you should go to the $y$ axis first, but it looks likely.

score 3 · Answer 3 · edited Jun 28 '19 at 04:29

3

All the other answers here are not good enough. The correct way to do it is as follows:

Reflect $P(3,5)$ about the $y$ axis to obtain $P'$ and reflect $Q(8,2)$ about the $x$ axis to obtain $Q'$.
Now consider points $R$ and $S$ on $y$ and $x$ axes respectively.
Then path length$= PR+RS+SQ= P'R+RS+SQ'$. So, this always represents a path from $P'$ to $Q'$.
To minimize this path length, a straight line joining $P'$ and $Q'$ would suffice.
Hence you can determine the coordinates of $R$ and $S$.
However, take another case whereby you reflect $P$ about the $x$ axis and $Q$ about the $y$ axis.

Follow the above steps and calculate the path length.

Compare the two lengths and consider the minimum length.

edited Jun 28 '19 at 04:29

nmasanta

9,222

answered Jun 28 '19 at 00:24

aditya gupta

370

Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, use MathJax. – dantopa Jun 28 '19 at 00:55
1

i am surprised people are downvoting this answer, even though this is the most ingenious answer there is. – aditya gupta Jul 03 '19 at 05:56

score 2 · Answer 4 · answered Feb 10 '13 at 18:46

You follow path of light.

First point is (3,5), then comes (0,a) and (b,0) and finally (8,2).
Line equation for the first segment is $y=\frac{5-a}{3}x+a$.
Line equation for the second segment is $y=-\frac{a}{b}x+a$.
Line equation for the third (last) segment is $y=\frac{2}{8-b}x-\frac{2b}{8-b}$.

Because it's reflection in both cases, $k_1=-k_2$ and $k_2=-k_3$:
1) $\frac{5-a}{3}=-\left( -\frac{a}{b}\right)$
2) $-\frac{a}{b}=-\frac{2}{8-b}$

Solve it!

Better see answer of Mark Bennet. – zaa Feb 10 '13 at 19:02 — zaa, Feb 10 '13 at 19:02

Find the length of the shortest path from the point (3,5) to the point (8,2) that touches the x-axis and also touches the y-axis

4 Answers4