## GRE 8767# 36

Forum for the GRE subject test in mathematics.
shalale
Posts: 2
Joined: Mon Sep 29, 2014 2:45 am

### GRE 8767# 36

Hi freinds;
can someone give me the solution of #36 on GRE 8767?
I know how I should solve it but I cannot get the answer.
thank you

Austin
Posts: 26
Joined: Fri Jun 01, 2012 9:32 am

### Re: GRE 8767# 36

We're trying to minimize the function $d(x,y)=\sqrt{x^2+y^2}$, subject to the restriction $xy=8$, which should immediately suggest the Lagrange multipliers method.

Of course, it's easier to work with $D(x,y)=x^2+y^2$. We can minimize this function, then take a square root. Then, letting $g(x,y)=xy$, we set $\nabla D = \langle 2x,2y \rangle = \lambda\nabla g = \lambda\langle x,y \rangle$, so we have $2x &= \lambda y,~~~~
2y &= \lambda x,~~~~
8 &= xy.$

We can quickly work out that the first two equations are satisfied when either $y=x$ or $y=-x$. But $y=-x$ can never satisfy $8=xy$, so we must have $y=x$, and thus $8=x^2$ and $8=y^2$, so $D(x,y)=16$, and $d(x,y)=4$.

This is the "this is a timed test and we're in a hurry" version. I'm sure someone else could give a more lucid explanation.