Question

The shortest distance from the curve xy=4 to the origin is ?

Answer 1

nice. where are you starting from?

Answer 2

i mean geometry or algebra 2 or whatever

Answer 3

You use the pythagorean theorem

Then find the derivative

Answer 4

\[x = \frac{4}{y}\]

\[\sqrt{(\frac{4}{y})^2 + y^2}\]

Answer 5

find the derivative of that then set it equal to 0.

y = 2 or -2

x = 2 or -2

2^2 + 2^2 = 4 + 4 = 8

sqrt(8) = 2 sqrt(2)

Answer 6

Calculus

Answer 7

Oops - this is Geometry:
Since the function is symmetric about the line y = x, distances come in matched pairs at P1 = (x1,y1) and P2 = (y1,x1). Evidently if there is a unique shortest distance, it has to happen at x = y (=2).

The line x+y=4 reaches its closest point to the origin at (2,2). Since the line x+y=4 lies entirely underneath your curve except at (2,2), your curve is everywhere at least that far from the origin, proving that (2,2) does indeed provide the global minimum distance, (2sqrt(2)).