A line through the point (2,2) cuts the x and y axes at A and B. Find the minimum length of the segment AB.

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- Apr 13th 2010, 01:19 PMrobertosavinOptimization Segment Minimum
A line through the point (2,2) cuts the x and y axes at A and B. Find the minimum length of the segment AB.

- Apr 13th 2010, 04:07 PMzzzoak
OA=2+x x-distance from x=2 to the right

tg a = 2/x

OB=(2+x)tg a=2(2+x)/x

f(x)=

f'(x)=0

OA=4 (x)

OB=4 (y).

|AB| min= - Apr 13th 2010, 04:28 PMArchie Meade
The line has a negative slope.

If it had a positive slope, it could go through the origin making the distance

between A and B zero.

If you draw the line going through (2,2) with a negative slope,

we have similar triangles by drawing lines from (2,2) to the x and y axes.

hence

let x-2=c, y-2=k

the length of the line segment is, using Pythagoras' theorem

You could differentiate this with respect to c and set the result = 0

to find the value of x causing the segment length to be a minimum.

Alternatively,

the segment length is

we can differentiate wrt the angle to find the minimum length

this occurs when the angle is 45 degrees

hence the minimum segment length is

or