calculus help amigos
posted by AMELIA
Find the point on the line 5x+5y+7=0
which is closest to the point (3,−4)

Reiny
Just use your "closest distance from a given point to a line" formula
D =  5(3) + 5(4) + 7 /√(5^2 + 5^2))
= 2/√50
= 2/5√2
= 2/5√2 units or √2/5 after rationalizing.
or , (the long and traditional way):
you know the slope of the given line is 1
so the slope of a perpendicular from (3,4) is +1
equation of that perpendicular is:
y+4 = 1(x3)
y = x  7
sub into the given line to find the intersection:
5x + 5(x7) + 7 0
5x + 5x  35 + 7 = 0
10x = 28
x = 2.8
y = 2.87 = 4.2
distance between (3,4) and (2.8, 4.2)
= √( (32.8)^2 + (4+4.2)^2)
= √(.04+.04)
= √.08 = √(8/100)
= √8/10
= 2√2/10 = √2/5, same as above
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