SOLUTION: Find a number t such that the distance between (-2,2) and (3t,2t) is as small as possible.

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Question 907680: Find a number t such that the distance between (-2,2) and (3t,2t) is as small as possible.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Use the distance formula.
D%5E2=%28-2-3t%29%5E2%2B%282-2t%29%5E2
D%5E2=%289t%5E2%2B12t%2B4%29%2B%284t%5E2-8t%2B4%29
D%5E2=13t%5E2%2B4t%2B8
To minimize the distance squared, take the derivative of the distance squared and set it equal to zero.
d%28D%5E2%29%2Fdt=26t%2B4%7D%7D%0D%0A%7B%7B%7B26t%2B4=0
26t=-4
t=-4%2F26
t=-2%2F13

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You could also do it geometrically.
(3t,2t) defines a line through the origin where y=%282%2F3%29x
You could find the line perpendicular to that line through the point (-2,2).
Perpendicular lines have negative reciprocal slopes so the slope would be m=-3%2F2
Using the point slope form of a line,
y-2=-%283%2F2%29%28x-%28-2%29%29
y-2=-%283%2F2%29x-3
y=-%283%2F2%29x-1
So then finding the intersection of the two lines,
%282%2F3%29x=-%283%2F2%29x-1
x%282%2F3%2B3%2F2%29=-1
x%2813%2F6%29=-1
x=-6%2F13
So then,
3t=-6%2F13
t=-2%2F13