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Geometry Level 4

A man starts from the point P ( − 3 , 4 ) P(-3,4) . He reaches Q ( 0 , 1 ) Q(0,1) but by touching the x x -axis at point R ( α , 0 ) (\alpha ,0) . What is the minimum distance that he can cover?

If the distance can be expressed as a a , Find a 2 a^2

This problem is original.


The answer is 34.

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Mehul Arora by Mehul Arora , 20, India

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4 solutions

Daniel Liu
Jul 12, 2015

Reflect his path from R ( α , 0 ) R(\alpha, 0) to Q ( 0 , 1 ) Q(0,1) across the x-axis. This gets rid of the "touch the x-axis" condition and just asks us to draw a path from P ( − 3 , 4 ) P(-3,4) to Q ′ ( 0 , − 1 ) Q'(0,-1) . But the shortest path clearly has length a = ( − 3 − 0 ) 2 + ( 4 − ( − 1 ) ) 2 = 34 a=\sqrt{(-3-0)^2+(4-(-1))^2}=\sqrt{34} so the answer is a 2 = 34 a^2=\boxed{34}

14 Helpful 0 Interesting 0 Brilliant 0 Confused

Sorry, I am a bit inactive these days.

Anyway, Thanks for the solution @Daniel Liu

Mehul Arora - 5 years, 11 months ago

How can we change the identity of a point? If Q is (0,1) why can we consider Q as (0,-1).

Ahmed Ali - 5 years, 10 months ago

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The point is reflected by the x-axis.

It's distance doesn't change that way :)

Mehul Arora - 5 years, 10 months ago
Aaghaz Mahajan
Mar 9, 2018

A nice problem!!!
But, I solved it by differentiation, longer but at least now I have come to know that reflection is an Awesome technique!!!

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Put a mirror on x_axis with reflating side to wards (-3,4). Its image will be at (-3,-4).
St. line from (-3,-4) to (0,1) is the shortest distance. So a= { 0 − ( − 3 ) } 2 + { 1 − ( − 4 ) } 2 S o a 2 = 34. \sqrt{\{0- (-3)\}^2+\{1-(-4)\}^2}\\ So \ a^2=34.

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Prabhav Bansal
Jul 19, 2015

It is not good as of daniel but yes, according to me it is a nice solution. Take the reflection of point Q taking x-axis as a mirror and mark the point as T. Its coordinates will be (0,-1). If we will join T and P, the intersection of TP at x-axis will be R. Now we can calculate the coordinates of R. it comes out to be R(-0.6,0). Now the rest of the solution is easy.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Your approach is identical to Daniel's.

Note that we didn't need to find the point R R , and could have calculated the distance T P TP directly.

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