Find the point (x0, y0) on the line 12x+12y=2 that is closest to the origin.

Question

I set ∇f to ∇g to get:
<2x,2y> = λ<12,12>
I then found λ = x/6 and λ = y/6, meaning that x = y
Once I plugged that back into the constraint, I found the answer to be:(1/12,1/12)
However, my online homework is saying my answer's incorrect. Did I do something wrong?

Answer 1

First reduce and change 12x+12y=2 to
6x + 6y = 1

Doing it another way:

slope of this line is -1
so the slope of the perpendicular is +1
and the line passing through the origin with slope 6 is y = x
sub into the given line 6x + 6y = 1
6x + 6x = 1
x = 1/12
y = 1/12
check is (1/12,1/12) on 12x+12y=2 ?
LS = 12(1/12) + 12(1/12)
= 1+1 = 2 = RS

My answer and your answer is correct, the book is wrong.