Optimizing a line segment

[Guest]

Problem: 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.

This one has me completely stumped! Any help is appreciated, a full solution would be just what the doctor ordered, though. :lol:

 

[pka]

Suppose that the line has (a,0) as the x-intercept and (0,b) as the y-intercept.
Then the equation of the line is \(\displaystyle \frac{x}{a} + \frac{y}{b} = 1\).
Because (2,2) is on the line we can get \(\displaystyle a = \frac{{2b}}{{b - 2}}\) and the length of the segment is \(\displaystyle \sqrt {\left( {\frac{{2b}}{{b - 2}}} \right)^2 + b^2 }\).

You should be able to maximize the length
I found b=4