find all points on the hyperbola y=1/x, that are closest to the origin on (1/2,2)

Answer 1

Let P(x,y) be the closest point
then if D is the distance, then
D^2 = x^2 + y^2
= x^2 + 1/x^2
2D dD/dx = 2x - 2/x^3
= 0 for a min of D
2x = 2/x^3
x^4 = 1
x = ± 1

if x=1, then y = 1
if x = -1, then y = -1

The two points closest to the origin are (1,1) and (-1,-1)

Where does the (1/2,2) come in?
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